Corporate Growth Dynamics under Hidden
Competition∗
Elmar Lukas† and Paulo J. Pereira§
† Faculty
of Economics and Management, Otto-von-Guericke-University, Magdeburg,
Germany
§ CEF.UP
and Faculdade de Economia, Universidade do Porto, Portugal.
Abstract
In this paper we extend the literature on dynamic corporate growth by including
the hidden competition environment, which could be appropriate to understand the
growth dynamics of industries in their early days. We derive the model that supports
the decisions regarding the timing, the choice of the best growth alternative (greenfield
investment or acquisition), as well as the optimal scale of the project. In contrast to
recent literature our numerical results reveal that it is not always optimal to speed up
market entry once a hidden competitor materializes. Rather, situations might occur
that motivate to further postpone market entry should a hidden competitor enter.
Moreover, high synergies among the competing firms induces the first mover to invest
less while the opposite holds for markets characterized by lower synergy potentials.
Finally, we show how our model can be used to predict the future constellation of
young industries once they mature and present new testable hypotheses.
Keywords: Real Options; Hidden Competition; Acquisitions; Greenfield Investment.
JEL codes: C73; D43; D81; D92; G31.
∗
We thank the Alcino Azevedo, Artur Rodrigues, Michael Flanagan, and conference participants at the
2016 Real Options Conference in Oslo. All remaining errors are of our own. Paulo J. Pereira acknowledges
the supported of FCT - Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and
Technology) within the project UID/ECO/04105/2013.
Corporate Growth Dynamics under Hidden
Competition
1
Introduction
Corporate growth may occur either via internal (organic) investments or via mergers and
acquisitions. In the former, firms increase their assets by investing in new facilities and
equipments, by hiring and training new employees, putting together all the resources in
order to exploit a profitable opportunity; whereas in the latter, firms’ growth is achieved by
acquiring and integrating the assets of an existing company, already acting in the industry.
In 2012, 18 months after being launch, Facebook purchased the photo-sharing network Instagram for $1 billion; in 2014, Google bought London-based artificial intelligence
company DeepMind for more than $500 million; and in 2002, eBay acquired the on-line
payments company PayPal in a deal valued at $1.5 billion. In all these examples, the
decision taken by the firms was to buy the competitor, instead of investing in the same
business and compete with him. However, a different approach was followed by IBM
when decided to keep developing its own personal computer (IBM PC), instead of trying
to acquire the young Apple Computer Inc at the time Macintosh (with the revolutionary
graphic interface) was launched. Also, Sony decided to invest in the video game industry
competing with the incumbents (Nintendo and Sega Systems), instead of buying one of
the player.
The decision to engage in one form of corporate growth instead of the other is crucial
for firms, and the dynamics of this choice already appears in the literature. McCardle and
Viswanathan (1994) develop a Cournot model for an oligopolistic industry where a new
potential entrant can move into the market by investing directly or, alternatively, through
the acquisition of an incumbent firm. Recently, Margsiri et al. (2008) follow a real options
approach to compare the opportunity to grow internally with the alternative growth by
acquisitions, and Perez-Saiz (2015) estimate a model of entry in a given industry considering the two alternatives. The determinants about the entry mode in entrepreneurship
1
context are presented by Parker and van Praag (2012) and Block et al. (2013).
A fundamental difference between our paper and the existing literature regards the
market context in which the decision, about investing or acquiring, needs to be made. In
fact, instead of assuming the business already exists and firms are active in the market,
we consider a different setting, where the market is just a potential business (without
any active firm) and where latent competitors remain idle and hidden, not disclosing any
information about their interest to enter the market. In this context, the decision to be
taken is a fundamental trade-off between a greenfield investment and a future acquisition
of a firm that may enter the market in the meantime, but that today remains unknown.
This setting is typically the case of industries in their early days, mainly when operating in the highly dynamic and disruptive markets. Companies like Uber, Tweeter,
WhatsApp (acquired by Facebook for $22 billion), or Viber (acquired by Rakuten for near
$1 billion), create a multi-billion dollar businesses coming to light unexpectedly, with no
prior information disclosure about their purposes.
Consistent with this setting, our model waves the assumption that commonly appears
in real options literature where full information is available for the competitors, allowing
them to endogenize the competitive behavior dynamics (e.g. Smets 1991, Grenadier 1996,
Shackleton et al. 2004, Pawlina and Kort 2006, Bouis et al. 2009, among many others).1
Some exceptions to this literature are Lambrecht and Perraudin (2003), that considers incomplete information about the competitor’s investment cost, Hsu and Lambrecht
(2007), that builds on Lambrecht and Perraudin (2003) in the context of patent racing,
and Nishihara and Fukushima (2008) where a start-up firm has incomplete information
about the behavior of a large competitor.
In a more extreme setting firms can act in a total hidden competition environment (see
Armada et al. 2011, Pereira and Armada 2013, Lavrutich et al. 2016) where the potential
competitors remain hidden, not unveiling their intention to enter the market, until the
moment they decide to do so. In this context, the threat comes not from known competitors, but, instead, from ”three guys in a garage” developing new, sometimes radical,
1
Refer to Azevedo and Paxson (2014) and Chevalier-Roignant et al. (2011) for an extensive review of
the main developments in dynamic real options games over the last two decades.
2
business ideas.
Many examples can be given in different industries: “(...) before its legendary rise, Apple was just three guys in a garage in Los Altos, California”2 , “Three Guys in a Garage Are
Turning Your Eyes Into Powerful Remote Controls”3 , “WestconGroup: From Garage Guys
To Global Distribution Powerhouse”4 . Particularly meaningful is Michael Bloomberg’s
statement: “My great fear is that there are three guys in a garage right now doing the
same thing to us that we did to Reuters and Dow Jones” (Ottoo 2000). Common to
these examples is the fact that, in many situations, competition can come apparently
from nowhere, starred by someone not yet known. In this competitive context, a company
willing to invest needs to account for the possibility of being preempted by a hidden rival,
without the possibility to endogenize his behavior. In fact, the entrance of a hidden competitor is essentially an exogenous event with some probability of occurrence, for which
there is no control measure other than preempt the hidden rival.
Assuming the market accommodates a finite number of active firms, the entrance of a
hidden rival leads to sudden decrease of the available places, which, in turn, may speed up
a firm’s investment. In this setting, the investment opportunity is approaching maturity as
new competitors enter into the market and disappears when the last place gets occupied.
The firm optimal behavior should balance the benefits of waiting and the risks of being
preempted by a hidden rival. In other words, in every moment in time, the company needs
to decide either to invest or to postpone the entrance into the market.
However, we realize in this paper that another alternative is available to a firm: the
option to acquire the hidden rival after his appearance in the market. This possibility
introduces more flexibility in the corporate growth process, and impacts the decision.
Acquiring the rival allows not only to eliminate any further risk arising from the hidden
competition, but also to benefit from the potential synergies. According to this idea,
it could be beneficial for a firm to acquire the hidden rival after his appearance in the
market, even when more places remain available for greenfield investment. The conditions
that make one alternative preferable to the other (greenfield investment preferable to an
2
3
4
http://www.businessinsider.com/history-of-apple-in-photos-2015-8
https://pando.com/2012/10/05/three-guys-in-a-garage-are-turning-your-eyes-into-powerful-remote-controls/
http://www.crn.com/slide-shows/managed-services/300077946/westcongroup-from-garage-guys-to-global-distribution-powerhouse.htm
3
acquisition, or vice versa) under the context of hidden competition, is the main focus of
this paper.
Besides connecting the dynamics of corporate growth and hidden competition, this
paper is also related to the literature concerning investment size (or capacity choice), in
particular in the context of uncertainty. Typically, the real options literature deals with
the optimal investment timing for a given pre-determined fixed capacity, but, in practice,
investors hold flexibility about dimension of the project. Only few models consider the
joint decision about investment timing and scale in a dynamic real options framework.
Dangl (1999) addresses this problem in a monopolistic context, concluding that when
uncertainty is high firm will choose to invest later in a larger scale. Huisman and Kort
(2015) extends discussion to a duopolistic setting, where firms need to determine their
timing of investment along with the optimal capacity level. The strategies about the
entry deterrence/accommodation and the effect of uncertainty on each player’s decision are
analyzed in detail. Other related contributions appear in Hagspiel et al. (2012), Boonman
and Hagspiel (2014), and Lavrutich et al. (2016).5
In our paper the choice about capacity is analyzed in a deeper manner. As we will see,
the scale of the investment will impact not only the timing but also the decision the firm
will take regarding the alternatives between the greenfield investment and the acquisition.
Under this setting, the capacity will be connected to the market share captured by the
firm, and the optimal investment (and so the optimal market share) will be endogenously
determined.
This paper contributes to the literature in several ways. Firstly, we extend the literature on dynamic corporate growth by including the hidden competition environment,
which could be appropriate to understand the growth dynamics of industries in their early
days. Secondly, we derive the model that supports the decisions regarding the timing, the
choice of the best alternative (greenfield investment or acquisition), as well as the optimal
size for the project. Thirdly, through a numerical example, we introduce and analyze
the conditions under which one growth alternative is better than the other, which can be
5
Please refer to Huberts et al. (2015) for a survey on the topic.
4
useful for supporting the decisions in real world. In addition, several testable hypothesis
are set. Finally, we present a computational experiment and we observe which future
industry constellations are most likely observed under different scenarios settings, namely
for different levels of synergies.
The paper unfolds as follows: Section 2 presents in detail the derivation of the model
considering the different stages in the market, as well the corresponding optimal alternatives; Section 3 analyses the results based on a numerical example and suggest several
empirical implications, while Section 4 highlights the path-dependent evolution of an industry by means of Monte Carlo simulation and Section 5 concludes.
2
The Model
Consider a firm facing the chance to enter in a market where only n companies can be
placed.6 For the sake of convenience, assume the market is a duopoly, and so n = 2.
The firm acts in a hidden competition environment, where potential competitors remain
unrevealed until one observes their entrance into the market. Given the places available,
three stages can be considered: the first stage when no company is active in the market,
and so the two places remain available (n = 2), then when a company takes a place in the
market and only one more place is available (n = 1), and finally, the last stage, when two
companies are in and there is no available room for more players (n = 0).
In each of these stages the firm has different options to consider. When there is no
company in the market the firm has to decide either to launch the project or to wait,
facing, in the latter case, the risk of an sudden entrance of a hidden rival. If an entrance
occurs while waiting to invest, the firm maintains the same options as in previous stage
(invest, occupying the last place in the market, or wait), but a new alternative arises: the
option to acquire the rival company. If the firm continues to wait and a second hidden
rival enters the market, the opportunity for greenfield disappears but remains the option
to acquire the hidden entrant.7 All the set of options the firm hold in each stage are
6
The derivation for an investment-only setting can be found in Pereira and Armada (2013).
For simplicity, we assume the firm only considers the acquisition of the first hidden rival entering the
market.
7
5
resumed in Figure 1.
Greenfield investment
or
wait
Acquire the rival,
or
greenfield investment
or
wait
No places available
n=2
n=1
n=0
Figure 1: The options available to the firm in each stage of the market, where n stands
for the places available in the market.
Let Fn (X) represent the value of the investment opportunity as a function of the
available places in the market (n) and the present value of the expected cash flows of
whole market (X). We assume that X follows a geometric brownian motion process:
dX = αXdt + σXdW
(1)
where X > 0, and α = r − δ represents the expected drift (r is the risk free rate and
δ is the dividend-yield), σ the instantaneous volatility, dW is the increment of a Wiener
process. Risk-neutrality is assumed.
Additionally, we assume that n ∈ {0, 1, 2} follows a Poisson process:
dn = −dq
(2)
where
dq =



1 with probability λn dt
(3)


0 with probability 1 − λn dt
The increment dq corresponds to a decrease in the available places in the market due
to an entrance of a hidden rival. The parameter λn is the mean arrival rate during the
period of time dt. We assume that λn increases as n decrease, since it would be more
6
likely the entrance of a hidden rival when the places become more scarce, and so λ2 < λ1 .
Regarding the places in the market, a first condition can be set:
F0 (X) = 0
(4)
which states that if there is no available room for more companies in the market, the
option to invest should be worthless.
Based on the standard arguments, Fn (X) must satisfy the following ordinary differential equation (ODE), in the continuation region:
1 2 2 00
σ V Fn (X) + αFn0 (X) − rFn (x) + λn [Fn−1 (X) − Fn (X)] = 0
2
(5)
where the last term captures the expected loss due to the entrance of a rival.
The solution for Fn (X) depends on Fn−1 (X) and we need to move backwards in order
to find it. Remember the value in the very last stage is given by Equation (4), and so we
start at n = 1.
2.1
The decision when one place remains available
Let us assume that one hidden rival is already placed in the market, capturing a market
share s and only one more place is available in the market. Naturally, if the active firm
captures the share s, the other firm will get the remaining (1 − s). As will see later on,
the optimal market share, s, will be endogenously determined in our model. One may
say that, in a context of hidden competition, the market share for the hidden rival is, by
definition, unknown ex-ante. However, we can argue that in such a context the best guess
is to assume the market share for the hidden rival is equal to the one the firm would choose
if entering first in the market.
Since one more place is available the firm needs to choose among the following three
alternatives: the greenfield investment, the acquisition of the rival now placed in the
market or the maintenance of both options, by deferring the decision. Let as start with
the alternative for greenfield investment.
7
2.1.1
The value of the option to invest
When only one more place is available in the market, Equation (5) can be re-arranged as
follows:
1 2 2 00
σ x F1 (X) + αxF10 (X) − (r + λ1 )F1 (X) = −λ1 F0 (X)
2
(6)
Accounting for the condition presented in Equation (4), the right-hand side equals zero
and the general solution is well known:
F1 (X) = A1 X β1 + A2 X β2
(7)
where β1 and β2 are, respectively, the positive and the negative roots of the characteristic
equation 0.5σ 2 β(β − 1) + αβ − (r + λ1 ) = 0, and A1 and A2 are arbitrary constants that
need to be determined.
Given the fact that the project must be worthless if not producing any cash flows, i.e.:
Fn (0) = 0
(8)
the constant A2 must be set equal to zero. The other constant A1 can be found along
with the optimal investment trigger (X1∗ ), using the value-matching and smooth pasting
conditions, respectivelly:
F1 (X1∗ ) = X1∗ (1 − s) − I(1−s)
F10 (X1∗ ) = 1 − s
(9)
(10)
where I(1−s) stands for the investment needed for capturing the market share (1 − s).
Equation (9) sets the Net Present Value when the investment is optimal, and Equation
(10) ensures the function is continuously differentiable along X.
8
Solving the equations we get:

X β1

∗

 X (1 − s) − I(1−s)
X1∗
F1 (X) =


X(1 − s) − I
(1−s)
where
1
α
β1 = − 2 +
2 σ
s
α
1
−
2
σ
2
2
+
if X < X ∗
(11)
if X ≥
X∗
2(r + λ1 )
>1
σ2
(12)
and the optimal trigger to invest is:
X1∗ =
2.1.2
1
β1
I
β1 − 1 1 − s (1−s)
(13)
The value of the option to acquire the rival firm
Consider now the option to acquire the rival company that has entered the market. We
follow recent literature by assuming that the takeover is the result of a noncooperative
bargaining game (as in Lukas and Welling (2012)). In particular, while the firm offers
the incumbent a premium Ψ > 0 in exchange for some synergies ξ > 0 (as a percentage
of firm’s value) the target firm has to time the asset sale. Moreover, we assume that
transaction costs arise for both the target firm and the acquirer, i.e. εC and (1 − ε)C,
respectively, where ε ∈ (0, 1) indicates the fraction of the transaction costs assigned to the
target firm. Because the rival firm was hidden before, we will assume that he optimally
chooses the same market share when he had decided to enter the market, i.e s, previously.
Hence his timing decision to sell the target with respect to X̄1 (s) solves the following
optimization problem:
f1 (X) = max E ((Ψ − 1)Xs − εC) e−rτ ,
τ
X γ1
= (Ψ − 1)X̄1 s − εC
X̄1
(14)
(15)
where γ1 = β1 (λ1 = 0).
Consequently, the acquisition occurs once X hits an optimal trigger value from below,
9
i.e. X = X̄1 with:
X̄1 =
γ1
εC
γ1 − 1 (Ψ − 1) s
(16)
Obviously, the timing decision depends on the premium X̄1 (Ψ) offered by the acquiring
firm. As the acquirer is interested in maximizing his option value, the optimal premium
is the solution to the following optimization problem, i.e.:
max
Ψ
(ξ − Ψ)X̄1 s − (1 − ε)C
X
X̄1
γ1 (17)
It follows that the optimal premium Ψ∗ paid results to:
Ψ∗ = 1 +
2.1.3
ε(ξ − 1)(γ1 − 1)
ε + (γ1 − 1)
(18)
The decision between the two alternatives
Consider that stage n = 2 suddenly changes to stage n = 1, meaning that a hidden rival
has entered the market. In such a context, the firm needs to decide which alternative to
follow, acquisition of the rival or greenfield investment (occupying the last place in the
market). We assume the decision is triggered by the change in the market stage and simply
follows a value maximization rule:
Ω(X) = max[F1 (X), f1 (X)]
(19)
This means that the firm will decide for the alternative of most value, immediately
after the entrance of a hidden rival into the market. After identifying the best alternative,
the firm starts monitoring the state variable, exercising the option at the corresponding
optimal moment.
2.2
The decision when the two places remain available
Let us now move backwards to the initial stage, where no one is in the market. Based on
the generic differential equation presented in (5), and considering the rule stated in (19),
the value function F2 (x) must satisfy appropriate ODE presented below.
10
2.2.1
If greenfield investment is prefereble to an acquisitsion
Under the circumstances where greenfield investment is optimal at n = 1 (i.e., Ω(X) =
F1 (X)), the value function F2 (X) must satisfy the following ODE:

X β1

∗

−λ2 X1 (1 − s) − I(1−s)
1 2 2 00
X1∗
σ X F2 (X) + αXF20 (X) − (r + λ2 )F2 (X) =

2

−λ2 X(1 − s) − I
(1−s)
, X < X1∗
,X≥
Following the standard procedures, and for the current level of the state variable, X,
the value of the option to invest at the stage n = 2 becomes:
F2 (X) =
where c1 =
λ2
λ2 −λ1 ,

X β1

η
∗

1

b1 X + c1 X1 (1 − s) − I(1−s)


X1∗


b3 X η1 + b4 X η2 + c2 X(1 − s) − c3 I(1−s)






Xs − I
(s)
c2 =
λ2
r−α+λ2 ,
and c3 =
λ2
r+λ2 .
, X < X1∗
, X1∗ ≤ X < X2∗
(21)
, X ≥ X2∗
I(s) stands for the investment that is
required to capture the market share s. Additionally, η1 is similar to Equation (12) but
replacing λ1 by λ2 , and
η2 =
α
1
−
−
2 σ2
s
1
α
−
σ2 2
2
+
2(r + λ2 )
<0
σ2
(22)
The solutions for the unknowns b1 , b3 , b4 , and X2∗ are obtained by solving the following
set of equations:
b1 X1∗η1 + c1 X1∗ (1 − s) − I(1−s) = b3 X1∗η1 + b4 X1∗η2 + c2 X1∗ (1 − s) − c3 I(1−s)
η1 b1 X1∗η1 −1 + c1 (1 − s) = η1 b3 X1∗η1 −1 + η2 b4 X1∗η2 −1 + c2 (1 − s)
b3 X2∗η1 + b4 X2∗η2 + c2 X2∗ (1 − s) − c3 I(1−s) = X2∗ s − I(s)
η1 b3 X2∗η1 −1 + η2 b4 X2∗η2 −1 + c2 (1 − s) = s
which together ensure that F2 (X) is continuous and differentiable along x.
11
X1∗
(20)
The solutions presented above are appropriate for conditions that lead to X1∗ < X2∗
(i.e., the triggers for investing is smaller when only one place is available in the market).
However, if the greenfield is comparatively more attractive at n = 1 than at n = 2 (namely,
when a larger market share is captured by the firm in the later stage greenfield investment),
an inverted order of the triggers, X2∗ < X1∗ , may occur. For obtaining the solutions for
all range of s this inverted order must also be considered. In this case, the solutions for
F2 (X) and for X2∗ are as follows:

X β1

η
∗

1
b5 X + c1 X1 (1 − s) − I(1−s)
X1∗
F2 (X) =



Xs − I(s)
, X < X2∗
,X≥
(23)
X2∗
where
b5 =
X2∗ s
− I(s) −
c1 X1∗ (1
− s) − I(1−s)
X2∗
X1∗
β1 ! 1
X2∗
η1
(24)
and the trigger X2∗ is the numerical solution of the equation:
(η1 −
1)X2∗ s
− η1 I(s) + (β1 −
η1 )c1 X1∗ (1
− s) − I(1−s)
X2∗
X1∗
β1
=0
(25)
subject to X2∗ < X1∗ .
The inverted order of the triggers occurs whenever the market share s is between sa
and sb , the two roots of the following quadratic equation:
(η1 − 1)
2.2.2
s
β1 − η1 1 − s
β1
− η1
+
c1
=0
β1 − 1
1−s
β1 − 1
s
(26)
If the acquisition is preferable to greenfield investment
For the situations where the acquisition of the hidden rival is optimal (i.e., Ω(X) = f1 (X)),
the solutions regarding F2 (X) are as follows:
12



−λ
Ψ∗ )X̄1 s
− (1 − ε)C
2 ξ−
1 2 2 00
σ X F2 (X) + αXF20 (X) − (r + λ2 )F2 (X) =

2

−λ2 ((ξ − Ψ∗ )Xs − (1 − ε)C)
X
X̄1
γ1
, X < X̄1
(27)
, X ≥ X̄1
Accordingly, the value of the option to invest at the stage n = 2 is:

X γ1

η
∗

1

B1 X + (ξ − Ψ )X̄1 s − (1 − ε)C


X̄1


F2 (X) = B3 X η1 + B4 X η2 + c2 (ξ − Ψ∗ )X̄1 s − c3 (1 − ε)C






Xs − I
(s)
, X < X̄1
, X̄1 ≤ X < X̄2
(28)
, X ≥ X̄2
where c1 , c2 , c3 , and η1 are as previously presented.
The solutions for the unknowns B1 , B3 , B4 , and X̄2 , are obtained by solving the
equations:
B1 X̄1η1 + (ξ − Ψ∗ )X̄1 s − (1 − ε)C = B3 X̄1η1 + B4 X̄1η2 + c2 (ξ − Ψ∗ )X̄1 s − c3 (1 − ε)C
η1 B1 X̄1η1 −1 + (ξ − Ψ∗ )s = η1 B3 X̄1η1 −1 + η2 B4 X̄1η2 −1 + c2 (ξ − Ψ∗ )s
B3 X̄2η1 + B4 X̄2η2 + c2 (ξ − Ψ∗ )X̄2 s − c3 (1 − ε)C = X2∗ s − I(s)
η1 B3 X̄2η2 −1 + η2 B4 X̄2η2 −1 + c2 (ξ − Ψ∗ )s = s
The solutions resulting from the equations above lead to X̄1 < X̄2 (i.e., the trigger for
the M&A, in stage n = 1, is smaller than the triggers to launch the greenfield investment
at n = 2). However, as previously, for some market shares an inverted order in the triggers
occurs, which must be considered for computing the triggers for all set of s. Under these
circumstances, the value function F2 (X) is:

X γ1


B5 X η1 + (ξ − Ψ∗ )X̄1 s − (1 − ε)C
X̄1
F2 (X) =


Xs − I
(s)
13
, X < X̄2
(29)
, X ≥ X̄2
where:
∗
B5 = X̄2 s − I(s) − (ξ − Ψ )X̄1 s − (1 − ε)C
X̄2
X̄1
γ1 1
X̄2
η1
(30)
and the trigger X̄2 corresponds to the numerical solution of the equation:
∗
(η1 − 1)X̄2 s − η1 I(s) + (γ1 − η1 ) (ξ − Ψ )X̄1 s − (1 − ε)C
X̄2
X̄1
γ1
=0
(31)
subject to X̄2 < X̄1 .
The inverted order occurs whenever
s<
H
H + η1 z
(32)
where
γ1
εC
γ
εC
∗
H = (γ1 − η1 ) (ξ − Ψ )
− (1 − ε)C + (η1 − 1)
∗
∗
γ1 − 1 Ψ − 1
γ−1Ψ −1
2.3
(33)
The optimal market share
While we have derived the flexibility value for the overall firm strategy, the firm has not yet
chosen the optimal scale of the project, i.e. how much it will invest in order to capture a
certain market share. Naturally, the higher the scale (and so the market share) the higher
the investment that needs to be spent. In particular, we seek for a functional relation
between the market share, s ∈ (0, 1), and the investment, I, such that I 0 (s) > 0, I 00 (s) > 0
which means that capturing larger portions of the market becomes increasingly expensive
(due to economic inefficiencies, for instance). The following expression is used:8
I(s) =
s
z
1−s
(34)
where z is a scale parameter.
Based on this relation an optimal market share need to be chosen such that it maximizes
8
Naturally, I(1 − s) =
1−s
z
s
14
the overall value of the investment opportunity for the firm. Thus, we have:
max
(F2 (X, s∗ ))
∗
s
(35)
which leads the corresponding optimal investment scale:
I(s∗ ) ≡ I(s∗ )
3
(36)
Numerical Example
The case of low acquisition synergies
For the numerical example we use the following parameter values: X0 = 0.5, µ = 0.1,
σ = 0.25, δ = 0.05, λ1 = 0.5, λ2 = 0.05, C = 4, = 0.1, and for the cost function we use
I(s) = z ∗ (s/1 − s) with z = 9. Let us first assume that a hidden competitor has already
entered the market, i.e. N = 1 and that the synergies that the hidden competitor would
have generated are low, i.e. ξL = 1.03.
As Figure 2 (a) indicates the option value for performing the greenfield investment is
thus always greater than due to an M&A, irrespectively which market share s is chosen previously. As uncertainty increases, both strategies become more valuable, i.e. their option
values increase and we also observe that the difference between f1 (X0 , s) and F1 (X0 , s) is
smaller for low market shares.
To what extend are the investment thresholds X1∗ and X2∗ affected by varying market
size. The results indicate (refer to Figure 2(b)) that X1∗ monotonically decreases as the
market share s increases which is reflected by Equation (13) and the fact that I1−s =
z(1−s)/s. Consequently, the more market share the firm has missed to secure initially the
less market share it can attract once the hidden competitor enters the market. Obviously,
the low market share comes at a lower cost and as a result the optimal investment threshold
decreases as market share increases. Moreover, the usual result stands out, i.e. the higher
the cash flow uncertainty the higher the optimal investment threshold becomes. Finally,
the optimal investment threshold decreases as the threat of a new entrant increases in
15
n = 1, i.e. as λ1 increases. This result is in line with previous results and indicates
that the firm’s propensity to invest in the greenfield investment in the second stage will
increase as the probability of a new hidden competitor increases (see e.g. Pereira and
Armada (2013)).
Now let us assume that the firm is faced with the decision to enter the market by
means of a greenfield investment in n = 1. Ceteris paribus, the firm anticipates that a
greenfield investment in the subsequent stage is more favorable and due to the fact that
no hidden competitor has entered so far it will invest as soon as the cash flow X hits
the optimal investment threshold X2∗ from below. As Figure 3 (b) reveals this optimal
threshold increases as the market share increases. This is due to the fact that a higher
market share implies higher sunk cost which raise the opportunity cost of giving up the
option to invest. Obviously, the subsequent option value due to greenfield investment
when competition has materialized is also priced in which dampens the increase of X2∗ for
situations characterized by high sunk costs. Moreover, we find that the firm’s propensity
to invest will increase should the probability of a new entrant in either n = 1 or n = 2
increase, i.e. X2∗ will decrease as λ1 and λ2 increase.
While these results replicate previous ones, we, however, find that the choice of market
share renders the firm’s overall entry strategy significantly. As can be seen from Figure 2,
for high market shares we have X2∗ > X1∗ indicating that postponing investment N = 2 is
very attractive for the firm. However, it will speed up investment should a new competitor
enter the market. On the contrary, for low market shares situations might occur where the
firm will prolong waiting in reaction of the entry of a new competitor, i.e. when X1∗ > X2∗ .
Put different, the threat of competition will not lead to an increased propensity to speed
up investment.
Finally, Figure 2 (c) depicts how the choice of market share affects the overall option
value for the firm at N = 2, i.e. F (X0 , s). While an increase in cash flow volatility
increases the option value an increased probability that a new entrant enters either in
n = 1 or n = 2 reduces the option value. More importantly, however, the figure also
reveals that F (X0 , s) exhibits an inverted U-shape with respect to the market share s.
16
(a)
0.3
(b)
140
F 1(σ=0.25)
(c)
1.4
x 2*(s)
x 1*(s)
f 1(σ=0.25)
F 1(σ=0.35)
x 2*(s,σ=0.3)
x 1*(s,σ=0.3)
f 1(σ=0.35)
F 1(σ=0.40)
x 2*(s,λ 1=0.7)
x 1*(s,λ 1=0.7)
f 1(σ=0.40)
x 2*(s,λ 2=0.01)
120
0.25
1.2
100
0.2
1
0.15
F2 (s)
optimal thresholds
F1 (s), f1 (s)
80
0.8
60
0.1
0.6
40
0.05
0.4
20
F 2(s)
F 2(s,σ=0.36)
F 2(s, λ 2=0.01)
F 2(s, λ 1=0.7)
0
0.1
0.2
0.3
0.4
s
0.5
0.6
0.7
0
0.1
0.2
0.3
0.4
s
0.5
0.6
0.7
0.2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
s
Figure 2: Greenfield investment dominant market entry where (a) shows the option
values F1 (X0 , s) and f1 (X0 , s) as a function of market share s for different
uncertainty levels, (b) the critical investment thresholds X1∗ and X2∗ as a
function of market share s for different uncertainty levels, and (c) the option
value of the overall greenfield investment at n = 2, i.e. F2 (X0 , s) as a function
of market share s.
Consequently, an increased commitment to invest more in order to obtain a larger stake
in the market leads to a larger flexibility value. However, at a certain market share the
overall option value decreases indicating that it becomes less valuable to invest too much.
Obviously, an optimal market share exists for which the option value becomes maximal.
Solving for the optimal market share leads to the following results (see Figure 3(a)).
There is an ambiguous relationship between the optimal market share s∗ and uncertainty.
In particular, the results indicate that for low levels of uncertainty the optimal market
share will decrease as uncertainty increases. Contrary, for higher levels of cash flow uncertainty the firm will opt for a higher market share as uncertainty increases. If the threat
of a new entrant becomes more severe, i.e. λ1 increases the U-shape pattern remains,
17
however, the overall market share increases indicating that the firm will thus prefer to
choose a larger market share. Alike, an increase in λ2 will lead to higher (lower) levels
of optimal market shares when uncertainty is low (high). Regarding the impact of λ1 on
the propensity to invest at n = 1 and n = 2 we find mixed results. While an increase in
λ1 leads to an increased propensity to the greenfield investment once a hidden competitor
has already entered we find that the opposite is the case once a competitor has not yet
materialized at all (see Figure 3 (b) and (c)). Consequently, an increase of λ1 indicating
that the last seat is about to be taken earlier leads to an increase of X2∗ which is opposite to the results of previous literature (e.g. Armada et al. (2011), Pereira and Armada
(2013)). Obvious, in n = 2, the firm is also subject to the threat that a new competitor
enters the market. Here we find that an increase in λ2 will lead to a (lower) higher optimal
investment threshold X2∗ (X1∗ ) when uncertainty is low (high).
Finally, we find an ambiguous timing-scale effect. More precisely, when uncertainty is
generally high we find that an increase of cash flow uncertainty leads to higher investment
levels and larger optimal investment threshold indicating a positive timing-scale effect.
For low levels of cash flow uncertainty, however, an increase of σ leads to an decrease in
s∗ while X2∗ increases (Figure 3).
The case of high acquisition synergies
As synergies increases from ξL = 1.03 to ξH = 1.25, the M&A strategy becomes dominant
in n = 1, i.e. once a hidden competitor enters (see Figure 4). Hence, the firm favours
to acquire the new entrant instead of committing to organic growth. As opposed to the
previous greenfield investment where missing to secure s in n = 2 leads to having (1 − s) in
n = 1 the firm’s option to acquire the rival secures any previous committed market share.
Hence, acquiring the rival acts as some kind of natural hedge. A direct result is that the
option value assigned to acquiring the rival in n = 1 is no longer inverted U-shaped and
increases as market share increases. In addition, a higher market share lowers the optimal
acquisition threshold X̄1 . Obviously, as cash flow uncertainty increases the option value f1
and X̄1 increase, too replicating well known results. Apart from securing any previously
18
(a)
0.635
(b)
70
(c)
20
s(σ)
s(σ, λ 1=0.7)
s(σ, λ 2=0.055)
68
19.5
0.63
66
19
64
0.625
62
x *1 (σ)
x *2 (σ)
optimal share s *
18.5
60
18
0.62
58
56
0.615
17.5
54
17
52
x2G * (σ)
x2G * (σ, λ 1=0.7)
x1G * (σ)
x1G * (σ, λ 1=0.7)
*
x1G * (σ, λ 2=0.055)
x2G (σ, λ 2=0.055)
0.61
0.1
0.15
0.2
0.25
0.3
50
0.1
0.15
0.2
σ
0.25
0.3
σ
16.5
0.1
0.15
0.2
0.25
0.3
σ
Figure 3: Greenfield investment dominant market entry where (a) shows the optimal
market share s(σ) as a function of uncertainty, (b) shows the critical investment thresholds X2∗ as a function of uncertainty, and (c) shows the critical
investment thresholds X1∗ as a function of uncertainty.
fixed market share the additional advantage of committing to an M&A in n = 1 is that
it rules out the threat that the last seat might be taken in the market by another hidden
firm. Consequently, λ1 does not impact the decision making.
Turning to the overall option value when n = 2 we find that the option value is
again inverted U-shaped as market share increases. This effect is based on the same
forces as described previously, i.e. increasing cost to scale which drive the greenfield
investment decision. As expected an increase in s leads to a higher investment threshold
X̄2 . Moreover, we see from Figure 4 that an increase in cash flow uncertainty increases F2
and X̄2 , respectively.
Solving for the optimal market share leads to the following results (see Figure 4).
Firstly, as uncertainty increases the firm will opt for a higher market share in n = 2. In
19
(a)
0.8
(b)
700
F 1(σ=0.25)
f 1(σ=0.25)
F 1(σ=0.35)
x *1(s)
x 2*(s,σ=0.3)
f 1(σ=0.35)
F 1(σ=0.40)
0.7
(c)
1.4
x *2(s)
x *1(s,σ=0.3)
f 1(σ=0.40)
x 2*(s,λ 1=0.7)
600
1.2
0.6
500
1
0.4
400
F2 (s)
optimal thresholds
F1 (s), f1 (s)
0.5
0.8
300
0.3
0.6
200
0.2
0.4
100
0.1
F 2(s)
F 2(s,σ=0.36)
F 2(s, λ 2=0.01)
0
0.1
0.2
0.3
0.4
0.5
s
0.6
0.7
0.8
0
0.1
0.2
0.3
0.4
0.5
s
0.6
0.7
0.8
0.2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
s
Figure 4: M&A dominant market entry where (a) shows the option values F1 (X0 , s)
and f1 (X0 , s) as a function of market share s for different uncertainty levels,
(b) the critical investment thresholds x∗1 and x∗2 as a function of market
share s for different uncertainty levels, and (c) the option value of the overall
greenfield investment at n = 2, i.e. F2 (X0 , s) as a function of market share
s.
comparison to the alternative greenfield-greenfield entry strategy discussed before, we do
not find a U-shaped relationship and the results further indicate that the greenfield-M&A
entry strategy leads to smaller market shares and lower investment levels, respectively.
However, the possibility of securing the intended market share by means of a natural
hedge, i.e. having the option to buy the first-moving firm later on serves as an incentive
to invest earlier. As Figure 4 (b) reveals the optimal greenfield investment threshold X̄2 is
smaller than X2∗ . Obviously, as uncertainty increases the propensity to delay investment
increases, too.
In line with the literature we find that an increased threat of being preempted by a
hidden competitor increases the propensity to speed-up investment, i.e. X̄2 decreases. This
20
goes hand-in-hand with a lower optimal market share indicating lower levels of investment.
Interestingly, the optimal acquisition threshold X̄1 exhibits a U-shape pattern with respect
to uncertainty. Consequently, for low levels of cash flow uncertainty, an increase in σ will
increase the propensity to acquire the rival sooner while for higher levels of uncertainty
an increase in σ will have the opposite effect. Moreover, due to the fact that an increased
threat of being preempted in n = 2, i.e. a larger λ2 , leads to an adjustment of the optimal
market share we will see that such a threat also translates into an increased propensity to
delay the acquisition once the hidden competitor has materialized in n = 1 (see Figure 5).
(a)
0.7
(c)
65
(b)
100
s(σ)
s(σ, λ 2=0.055)
98
60
0.65
96
55
94
50
optimal share s *
0.6
x *1 (σ)
x *2 (σ)
92
45
90
0.55
40
88
35
86
0.5
30
84
x2MA * (σ)
x2MA * (σ, λ 2=0.055)
0.45
0.1
0.15
0.2
σ
0.25
0.3
25
0.1
0.15
0.2
σ
0.25
x1MA * (σ)
x1MA * (σ, λ 2=0.055)
0.3
82
0.1
0.15
0.2
0.25
0.3
σ
Figure 5: M&A dominant market entry where (a) shows the optimal market share s
as a function of uncertainty, (b) shows the critical investment thresholds X2∗
as a function of uncertainty, and (c) shows the critical investment thresholds
X1∗ as a function of uncertainty.
A final question remains which is linked to the interaction of optimal market share
and the impact of investment cost under uncertainty. In particular, we want to check
whether less expensive market entries lead to higher market power. From Equation (34) it
21
becomes apparent that z measures how strongly market share affects cost. Exemplary, for
low values of z an increase increase in market share leads to an overall smaller increase of
investment cost than for high values of z. Our results, however, indicate that the optimal
choice of market share is less sensitive to z. As Figure (6) indicates, the firm will in
general attempt to capture a market share larger than fifty percent. This is independent
from whether the firm prefers greenfield or M&A should a hidden competitor preemt the
firm. Moreover, even for very low values of z a favor for a majority market share sustains.
0.68
s *GF (σ)
0.66
s *GF (σ, z=11)
s *GF (σ, z=7)
0.64
s *MA (σ)
s *MA (σ, z=11)
optimal share s *
0.62
s *MA (σ, z=7)
0.6
0.58
0.56
0.54
0.52
0.5
0.48
0.15
0.2
0.25
0.3
σ
Figure 6: Optimal market share as a function of uncertainty and z.
Testable implications
From the aforementioned comparative-static analysis we can deduce the following hypothesis:
Hypothesis 1: For markets that are characterized by a high threat of new entrants in
both stages, i.e. n = 1, 2, which do not generate valuable synergies the first-moving firm
will favor a higher (lower) market share should the operate in highly volatile (less volatile)
industries. In such a setting, firm’s operating in less volatile industries have a tendency
22
to postpone rather to speed up greenfield investment as a sequence of hidden competition.
Hypothesis 2: For markets that are less volatile with strong synergy potential among
the firms exhibit a higher probability that an entrepreneurial firm invests soon followed by
a greater propensity of consolidation after the first-entrant has materialized.
Hypothesis 3: If there is a promising market for M&As due to strong synergies
potentials, then these industries exhibit lower levels of investment at the time the firstmoving firm materializes.
Hypothesis 4: In industries with less promising market for M&As, first-movers tend
to capture larger stakes of the market, independently from the level of the uncertainty.
4
Computational Experiment
In the following we will perform two computational experiments in order to analyze the
structure of an industry given that rivals are hidden. Especially, we are interested how
likely it is to observe a specific industry constellation, e.g. the two hidden competitors
will enter the market by means of greenfield while the firm acquires the first mover later
on. Against this background, we choose a time horizon of 30 years and calculate the firm’s
cash-flows X on a monthly basis by using discretizing the geometric Brownian Motion.
Subsequently, we ran a Monte-Carlo Simulation with 10.000 trials and count the number
of times the optimal investment thresholds were hit as well as how often one or two of
the hidden competitor had materialized. The following parameters are used: X0 = 10,
r = 0.1, δ = 0.05, σ = 0.2, C = 0.5, z = 5, = 0.5, λ2 = 0.1, λ1 = 0.2.
Suppose that the firm estimates that one hidden competitor will mimic its innovation
efforts within the next 10 years and that once the firm has invested this threat of competition will be doubled, i.e. the next competitor will most likely enter within the following five
years. Further, the synergies between the hidden competitor and the firm are estimated
at 1.1. Then, two major industry constellations are possible. First, it is very likely that
the first entrant will be the hidden competitor while the firm moves second (i.e. (1C,F)
approx. 62%). In such a setting, the first mover controls 59% of the market while the
second mover holds a 41% market share. Obviously, both firms will perform a greenfield
23
investment. Moreover, it is also very likely that both hidden competitors enter within the
30 years simulation period by means of a greenfield investment and the firm does not enter
because all seats are already taken (1C,2C). Such a scenario accounts for roughly 25% of
the simulations trials (Figure 7). The two minor constellations left are that the firm will
be the first mover followed by a hidden rival (7%) and that only one hidden rival is in the
market and no other has committed to enter the market (5%). Obviously, scenarios that
highlight an empty market as well as M&A activity are neglectable.
Jumps at: t1 =52 and t2 =208
50
1
Market Entry Sequence
Market Shares
x(t)
x g1
x g2
40
41%
x ma
gamma
Threshold
30
20
59%
10
0
0
1000
2000
Time
0
1C
F
2C
Timing
Number of Simulations: 10000
70
Likelihood in %
60
50
40
30
20
10
0
F, 1C
F
Empty Market 1C, F
1C, 2C
1C
1C+F, 2C
1C+F
Industry Composition
Figure 7: Industry constellation after 30 years given low synergies. (1C = first competitor, 2C=second competitor, F=firm, 1C+F=Acquisition of First Entrant).
If, however, the estimated synergies change to 1.3 then the future constellation of the
industry changes dramatically. As figure 8 reveals, it is very likely that a hidden competitor
will invest first in the market by means of a greenfield investment and is subsequently,
acquired by the firm who will move second (1C+F,2C). In particular, the market share
24
of the first investor is quite severe, i.e. 80%. In such a setting it is also very likely
that a second hidden competitor captures what is left, i.e. 20%. In total, this industry
constellation accounts for almost 80% of the simulation trials. As can further be deduced
from the figure, four minor scenarios are also possible. First, the industry after 30 years
will consist of both hidden competitors ((1C,2C), 12%) while the second most likely minor
scenario is that the firm will acquire the first-mover and no other rival will enter ((1C+F),
7%). Finally, only one hidden competitor will serve the market ((1C), 4%). A situation
where the firm moves first and competes against a competitor is in 2% of the simulation
trials.
Jumps at: t1 =208 and t 2 =1716
100
Market Entry Sequence
1
Market Shares
x(t)
x g1
x g2
80
20%
x ma
gamma
Threshold
60
40
20
80%
0
0
500
1000
1500
2000
0
1C+F
2C
Time
Timing
Number of Simulations: 10000
80
70
Likelihood in %
60
50
40
30
20
10
0
F, 1C
F
Empty Market
1C, F
1C, 2C
1C
1C+F, 2C
1C+F
Industry Composition
Figure 8: Industry constellation after 30 years given high synergies. (1C = first competitor, 2C=second competitor, F=firm, 1C+F=Acquisition of First Entrant).
25
5
Conclusions
In this paper we extended the literature on real options under hidden competition. In
addition to the decision of investing or waiting, we consider the realistic alternative of
acquiring the hidden rival, after his appearance in the market. Hence, our dynamic model
accounts for simultaneously determine the best optimal growth strategy, the optimal timing of market entry as well as the optimal size of investment.
In contrast to recent literature our results reveal that it is not always optimal to
speed up market entry once a hidden competitor materializes. Rather, situations might
occur that motivate to further postpone market entry should a hidden competitor enter.
Moreover, high synergies among the competing firms induces the first mover to invest less
while the opposite holds for markets characterized by lower synergy potentials. Finally,
we show how our model can be used to predict the future constellation of young industries
once they mature and present new testable hypotheses.
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