Discussion Papers In Economics
And BusinessHow should a startup respond to acquirers? A real
options analysis

Michi NISHIHARA
Discussion Paper 20-24

March 2021

Graduate School of Economics
Osaka University, Toyonaka, Osaka 560-0043, JAPANHow should a startup respond to acquirers? A real options analysis∗
Michi NISHIHARA†

Abstract
Many startups have recently opted for sellout to large firms with great market power and profit
margins. This paper attempts to understand and guide such startup sellout decisions by developinga model in which the firm reacts to random approaches of high- and low-type acquirers. Optimally,
the startup takes either a high-price strategy—accepting only a high-type acquirer in a good economic state— or a flexible strategy—accepting a high-type acquirer in an intermediate economic
state and either type in a good economic state— based on a tradeoﬀ between sellout pricing andtiming eﬃciency. With asymmetric information, where the acquirer types are unobservable, thestartup accepts a high-price acquisition more eagerly and a low-price acquisition more restrictively
to reduce the acquirer’s information rent. Then, asymmetric information increases the probabilityof a high-price sellout and delays the sellout. A model analysis shows that the market parameters
as well as anticipation of acquirers (i.e., the acquirers’ arrival rate, valuation, and transparency)
greatly aﬀect the sellout price, sellout timing, firm value, and stock price reaction.
JEL Classifications Code: G34; G13; D82.
Keywords: M&A; real options; selling process; liquidity; asymmetric information.

∗

The version on 8 March, 2021. The preliminary version was entitled as “Real options with illiquidity of exercise

opportunities.” The author thanks Norio Hibiki, Junichi Imai, Takashi Shibata, and Kazutoshi Yamazaki for helpfulcomments. This research was presented at the ISERD 335th International Conference on Accounting and Finance inTaipei, the 5th Stochastic Modeling Techniques and Data Analysis International Conference in Chania, and the 2018 and2021 Spring National Conferences of Operations Research Society of Japan. The author thanks the participants for helpful
feedbacks. This work was supported by the JSPS KAKENHI (Grant number 20K01769, JP17K01254, JP17H02547).
†
Corresponding Author. Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka
560-0043, Japan, E-mail: nishihara@econ.osaka-u.ac.jp, Phone: 81-6-6850-5242, Fax: 81-6-6850-5277

1

1Introduction

Over the last few decades, industry concentration has grown, increasing the gap between the profitability of giant firms and others (e.g., Grullon, Larkin, and Michaely (2019), Autor, Dorn, Katz,
Patterson, and Reenen (2020)). For instance, Amazon, Apple, Google, Facebook, and Microsoft,the “tech giants” in the information technology industry, have greatly increased their market power
and profit margins. In such circumstances, many startups would prefer to be acquired by large
firms, rather than developing themselves as independent firms through initial public oﬀerings andgrowth investments (see Poulsen and Stegemoller (2008), Gao, Ritter, and Zhu (2013)). For example, Instagram chose to be sold to Facebook in order to expand its business in 2012. This study
tries to understand and guide such startup sellout decisions by developing a real options model asfollows.1
Assume that a startup with stochastic cash flows has an option for sellout to a large acquirer.
The acquirer can pay a higher price than the startup’s standalone value because it can extract
more cash flows from the startup’s assets (e.g., synergies, market power, and economies of scopeand scale). The startup times the sellout considering the irreversibility and sunk costs of the process.
Moreover, the model assumes two novel features. The first is the illiquidity of sellout opportunities
stemming from frictions in the merger and acquisition (M&A) searching and matching process. I
assume that the sellout is feasible only at the acquirers’ arrival time modeled by Poisson jump
times.2 The second feature is the heterogeneity of acquirers. Two acquirer types (hereafter calledhigh and low types) are considered, where the high type extracts more cash flows from the acquired
assets than the low type.
In the model, I derive the startup’s sellout policy and firm value analytically. The results arepresented in two cases. In the first case, the startup accepts only a high type when the state variable(i.e., the cash flow level) lies above a certain threshold. This policy is called the high-price strategy.
The firm value and policy in the high-price strategy coincide with those in the benchmark modelwith only high types. The high-price strategy pursues the pricing rather than timing eﬃciency. In
fact, the startup forgoes low-price acquisitions even when the state variable is high. In the second
case, the sellout policy involves two thresholds (the high-price and low-price sellout thresholds).
The startup accepts only a high type when the state variable lies between the two thresholds, andaccepts either type when the state variable lies above the low-price sellout threshold. This policy iscalled the flexible strategy. The flexible strategy pursues the timing rather than pricing eﬃciency.
In fact, the startup chooses sellout for the high state variable even when the sellout price is low. In
the flexible strategy, both the sellout time and price are unpredictable; this is consistent with the1

The real options approach is now widely accepted as an alternative to the net present value method, especiallyin decisions involving uncertainty and irreversibility (e.g., see Dixit and Pindyck (1994) and Trigeorgis and Tsekrekos(2018)).
2
The model considers the sellout process through private negotiations, where potential bidders approach the target
sequentially. Firms are sold in their growth stage through private negotiations rather than auction (see Fidrmuc,
Roosenboom, Paap, and Teunissen (2012)).

2

empirical evidence of high uncertainty in M&A deals (see Cornett, Tanyeri, and Tehranian (2011)
and Wang (2018)).
The tradeoﬀ between the sellout pricing and timing eﬃciency determines whether the startup
takes the high-price or flexible strategies. Notably, I derive the explicit condition and show thatthe startup is more likely to take the high-price strategy when the high type’s arrival rate andgrowth rate of the state process are relatively high. A numerical analysis of the model yields thefollowing comparative statics results. With lower acquirers’ arrival rate, the startup accepts an
acquisition more eagerly, but this sellout policy does not fully oﬀset the decrease in liquidity of
sellout opportunities. In fact, a lower arrival rate delays the sellout and decreases the firm value.
Contrary to the well-known eﬀect of the growth rate, the low-price sellout threshold increases withhigher growth rate. The startup restricts a low-price acquisition more severely because a higher
growth rate increases the advantage of high-price sellout. The startup with higher growth rate andlower volatility is more likely to be acquired by a high type (i.e., firm with higher eﬃciency). To myknowledge, these are novel predictions that are yet to be tested in the empirical M&A literature.
I extend the baseline results to studies with asymmetric information, where startups are unaware of the acquirer types. The asymmetric information model is relevant when the acquirers
are less transparent. The startup with asymmetric information faces the risk of sellout to a hightype at a low price. Then, in contrast to the symmetric information case, the high-type acquirer
gains information rent in acquisition. This result is consistent with the empirical evidence of low
acquisition premiums of less transparent acquirers (e.g., Bargeron, Schlingemann, Stulz, and Zutter (2008), Battigalli, Chiarella, Gatti, and Orlando (2017), and Golubov and Xiong (2020)). The
startup decreases the high-price sellout threshold and increases the low-price sellout threshold to
reduce the information rent.3 Compared to the symmetric information case, this distortion increases
the high-price sellout probability but delays the sellout timing, which results in social ineﬃciency.
This social loss due to asymmetric information tends to be highest when asymmetric information
changes the startup strategy from the flexible strategy to the high-price strategy.
Furthermore, I examine the firm value jump at the sellout time. This can be interpreted asacquisition premium or target stock price reaction. A sellout leads to a positive jump in firm
value from the unpredictability of sellout timing (and pricing). This finding is consistent withthe empirical evidence of positive target stock price reactions (e.g., Huang and Walkling (1987),
Boone and Mulherin (2007), and Hackbarth and Morellec (2008)). In the flexible strategy case,the jump size is not monotonic relative to the state variable because the startup accepts a lowprice acquisition for a high state variable. Similarly, the jump size is not monotonic relative to thearrival rates because the sellout price can change with the arrival rates. These results show thatthe target’s stock price reaction depends on the interactions between the sellout timing and price.
Finally, I explain the contribution of this study to the related literature. From the methodolog3

This finding is consistent with the intertemporal price discrimination literature (e.g., Stokey (1979), Landsberger andMeilijson (1985), Nishihara and Shibata (2019)). In diﬀerent settings, these studies show that a seller delays low-pricesales to reduce the information rent to high types.

3ical perspective, I solve the optimal stopping problem constrained to Poisson jump times. Dupuis
and Wang (2002) first solves this type of constrained optimal stopping problem. By extending
Dupuis and Wang (2002)’s model, Hugonnier, Malamud, and Morellec (2015a), Hugonnier, Malamud, and Morellec (2015b), and Morellec, Valta, and Zhdanov (2015) study the illiquidity eﬀects offund-raising on investment, while Nishihara and Shibata (2021) investigate the illiquidity eﬀects ofsellout opportunities on bankruptcy. However, all these studies assume the homogeneity of illiquid
option exercise opportunities. To my knowledge, this is the first study to solve the problem with
two option types arriving randomly at diﬀerent arrival rates. I also clarify the relationship between
the homogeneous and heterogeneous problems by deriving the explicit condition under which a
two-type problem is reduced to a one-type problem. Note that no asymmetric information problem
occurs for the homogeneous problem. This study first solves the asymmetric information problem.
The study also contributes to the real options literature on M&A. Lambrecht (2004) develops
friendly merger and hostile takeover models, while Morellec and Zhdanov (2008) study the biddercompetition eﬀects on takeover and capital structure decisions. Lukas, Pereira, and Rodrigues(2019) show that serial acquisition is optimal in certain situations. As regards the M&A of startups,
Nishihara (2017) examines the startup choice between sellout and initial public oﬀerings, whereas
Ferreira and Pereira (2021) examine the entry and exit (trade sale) decisions of venture capitalists.
In these studies with no frictions in the M&A searching and matching process, the M&A timing ispredictable, and hence, there is no shock in the target firm value at the M&A time. In contrast, thisstudy shows that the illiquidity of sellout opportunities distorts the timing decision and leads to a
shock at the sellout time. Nishihara and Shibata (2021) examine how illiquid sellout opportunities
aﬀect the capital structure and default decisions, although they assume homogeneous acquirers. By
assuming the heterogeneity of acquirers, this study examines the timing as well as pricing decisions
and the eﬀects of asymmetric information.
The remainder of this paper is organized as follows. Section 2 introduces the model setup.
Sections 3.1 shows the benchmark model results with homogeneous acquirers. Sections 3.2 and3.3 present the main model results with heterogeneous acquirers under symmetric and asymmetric
information, respectively. Section 4 analyzes the results numerically and explains the implications.
Section 5 concludes the paper.

2

Model setup

Consider a startup (or an entrepreneur in a certain section, with technology, patent, etc.) operating
and generating continuous streams of cash flows X(t), where X(t) follows a geometric Brownian
motion,dX(t) = µX(t)dt + σX(t)dB(t) (t > 0),

X(0) = x;

here, B(t) denotes the standard Brownian motion defined in a filtered probability space, (Ω, F, P, {Ft }),and µ, σ(> 0) and x(> 0) are constants. The net present value (NPV) of cash flows at time 0 is
equal to π(x) = x/(r − µ).

4Infinitely many high- and low-type acquirers value the firm as aH π(X(t))−Ia and aL π(X(t))−Ia ,respectively, where aH > aL > 1 and Ia ≥ 0.4 Throughout this study, H and L stand for high andlow types, respectively, and ∆a = aH − aL . The assumption that ai > 1 indicates that acquirers
extract more cash flows from the business than the startup can. The synergies associated withacquisition can be included in the parameters ai (i = H, L), while Ia represents the sunk cost in
acquisition. High and low types approach the startup at Poisson jump times with arrival rates λH
and λL , respectively. For model tractability, I assume that the Poisson arrival processes and X(t)are independent.5
The startup pays the transaction cost Ib (> 0) when acquired. I denote the total costs by
I = Ia + Ib . When the startup is sold at price ai π(X(t)) − Ia , it gains ai π(X(t)) − I instead oflosing the future cash flows’ NPV π(X(t)). The total payoﬀ then becomes (ai − 1)π(X(t)) − I. I
assume that the startup is risk-neutral and chooses between accepting and rejecting an acquirer
to maximize the expected firm value. If the firm rejects an acquirer, it has to wait until anotheracquirer arrives.
This study examines two types of information structures. In Section 3.2, I explore the symmetric
information case, where the startup can observe the acquirer types. In Section 3.3, I explore theasymmetric information case, where the startup cannot observe the acquirer types. In reality,
asymmetric information matters more when the acquirers have fewer business relationships withthe startup. It also matters when the acquirers are unlisted and have lower transparency. In caseof symmetric information, the startup can be sold at the fair price ai π(X(t)) − Ia contingent on
the acquirer type i. In contrast, in case of asymmetric information, the startup cannot post a pricecontingent on the acquirer type. For simplicity, I assume that the startup makes a take-it-or-leave-it
price oﬀer. An acquirer would purchase the startup if the oﬀer price is not higher than its valuation,
and would leave it if the price is higher than its valuation. In case of asymmetric information, ahigh type can obtain information rent by purchasing the startup at the low price aL π(X(t)) − Ia(for details, see Section 3.3).
The model is based on the following empirical observations. Startups tend to sell their growing
businesses to large firms because large firms can extract more profits from them by virtue of theirsynergies, market power, economies of scale and scope, speed in bringing products to the market,
and so on (e.g., Poulsen and Stegemoller (2008), Gao, Ritter, and Zhu (2013), Grullon, Larkin, andMichaely (2019)).6 A startup in the growth stage is typically sold through private negotiations,with the potential bidders approaching the target firm sequentially, rather than through auctions
4

I can solve the problem with more than two types in the same manner. In that case, the number of startup strategies

become more than two, and hence, I cannot derive an explicit condition for each strategy. The main implications in thispaper will remain unchanged.
5
Hugonnier, Malamud, and Morellec (2015a), Hugonnier, Malamud, and Morellec (2015b), Morellec, Valta, andZhdanov (2015), and Nishihara and Shibata (2021) also assume that the economic state and Poisson processes are
independent. In fact, one cannot analytically derive any solution without independence.
6
For a well-known example, Instagram was sold to Facebook in 2012. The sellout process is private negotiation rather
than auction.

5

(see Fidrmuc, Roosenboom, Paap, and Teunissen (2012)). Business sellout transactions, unlike thesellout of financial assets, involve high transaction costs, irreversibility, illiquidity, and price uncertainty. The real options model captures the sunk cost and irreversibility in real asset trading. By
adopting the Poisson arrival processes of acquirers, the model captures the illiquidity (i.e., searching and matching frictions) of the sellout being feasible only when the firm meets a counterparty.7
The model incorporates the heterogeneity of acquirers and asymmetric information, to capture thetransaction price uncertainty.
Finally, I explain how the model diﬀers technically from previous M&A real options models. In
most previous studies, the acquisition is predictable because the timing is endogenously determinedby the acquirer, target firm, or both firms. For example, in the takeover models of Lambrecht(2004) and Morellec and Zhdanov (2008), the target can precisely predict the acquirer’s takeover
time and optimize the terms (i.e., the price). Hackbarth and Morellec (2008) investigate theasymmetric information between the firm insiders and outside investors, although the merger timing
is predictable to the target firm and acquirer managers. Unlike these papers, this study assumes
that the target startup optimizes the price but cannot predict the acquirers’ arrival time, modelingthe acquirers’ arrival as an unpredictable and exogenous event. This assumption holds true when
friction exists in the searching and matching process. In fact, some firms fail to reach an agreement
in M&A negotiations. This assumption is also plausible when the acquirers’ arrival timing isdetermined by factors unobservable to the target startup (e.g., acquirers’ competitive strategy,
business environment, and long-term plan). Although Nishihara and Shibata (2021) model theacquirers’ arrival as an unpredictable and exogenous event, their model lacks acquirer heterogeneity(i.e., price uncertainty). Many papers (e.g., Boone and Mulherin (2007), Cornett, Tanyeri, andTehranian (2011), and Wang (2018)) empirically show high uncertainty in M&A deals, and thisstudy is consistent with them.

3

Model Solutions

3.1

Homogeneous acquirers

As benchmark, this subsection considers a one-type acquirer arrival setup. I fix i ∈ {H, L}, andassume that λi > 0 and λj = 0, where j = {H, L} \ {i}. Technically, the firm’s optimal sellout
timing problem can be regarded as an optimal stopping problem constrained to Poisson jump times,
which is solved in Dupuis and Wang (2002). I denote the firm value when an acquirer is absent as
Vi (x). When an acquirer arrives, the firm value becomes max{ai π(X(t)) − I, Vi (X(t))}, because
the firm optimizes the choice between accepting and forgoing the acquirer. As proved in Dupuis
and Wang (2002), the optimal policy becomes a threshold policy, that is, inf{t ≥ 0 | X(t) ≥ xi },7

Rhodes-Kropf and Robinson (2008) examine an M&A market with search frictions similar to those in this study, but

in diﬀerent contexts.

6and Vi (x) satisfies the ordinary diﬀerential equations (ODEs)µxVi′ (x) + 0.5σ 2 x2 Vi′′ (x) + x = rVi (x) (0 < x < xi ),(1)µxVi′ (x) + 0.5σ 2 x2 Vi′′ (x) + λi (ai π(x) − I − Vi (x)) + x = rVi (x) (x > xi ),(2)

where λi (ai π(x) − I − Vi (x)) in (2) corresponds to the fact that Vi (x) changes to ai π(x) − I withprobability λi dt in infinitesimal time interval dt in the sellout region x > xi .8 Region x < xi is theoperation region, where the firm operates normally, whereas region x ≥ xi is the sellout region,
where the firm aims for sellout. State process X(t) can cross the threshold xi many times because
the startup may not meet an acquirer in the sellout region.
As shown in Dupuis and Wang (2002), Vi (x) is continuously diﬀerentiable at xi 9 and satisfies
boundary conditions
Vi (xi ) = ai π(xi ) − I,(3)lim Vi (x) = 0,

(4)

lim Vi (x)/x < ∞.

(5)

x→0

x→∞

Condition (3) means that the startup is indiﬀerent to whether to accept or reject a proposal at
X(t) = xi (i.e., the optimality of xi ). Condition (4) is a standard condition arising from the factthat if x goes to zero, the stochastic process X(t) will stay at zero, and hence, Vi (xi ) goes to 0 (e.g.,
Dixit and Pindyck (1994)). Condition (5) occurs because the firm value is bounded by ai π(x).
Throughout the paper, I define the notations√)2 2(r + y)
(µ
µ
+
−
0.5
(> 1),βy = 0.5 − 2 +
σ
σ2
σ2
√
)2 2(r + y)(µµγy = 0.5 − 2 −−0.5+(< 0),
σ
σ2
σ2

(6)
(7)

for an arbitrary parameter, y > 0, and simplify the notations by β = β0 and γ = γ0 for y = 0. I
PValso define the zero-NPV thresholds xN= (r − µ)I/(ai − 1) for i = H, L. By solving (1) and (2)iwith boundary conditions (3)–(5), I have the following proposition. For proof of the proposition,
see Appendix A.
Proposition 1 The firm value Vi (x) (i = H, L) is given by π(x) + A1,i xβ
Vi (x) =λi (ai − 1)π(x)λi I−+ A2,i xγλi
 π(x) +r + λi − µr + λi8(x < xi ),(x ≥ xi ),(8)

ODEs (1) and (2) are similar to those in the model with operating costs and temporary suspension, where the firm

can freely switch between operation and suspension (see Chapter 6.2 of Dixit and Pindyck (1994)).
9This is not the smooth pasting (i.e., optimality) condition in Dixit and Pindyck (1994), but similar to the piecewise
C 2 property of the value function in Morellec, Valta, and Zhdanov (2015). This condition is also used in Chapter 6.2 of
Dixit and Pindyck (1994). Technically, this property follows from Theorem 4.4.9 in Karatzas and Shreve (1998).

7

where the sellout threshold xi and coeﬃcients A1,i , A2,i are defined byPV(r + λi − µ)((β − γλi )r + βλi )xN
i,(r + λi )((β − γλi )(r − µ) + (β − 1)λi )(ai − 1)π(xi ) − I
A1,i =,xβi()
1(ai − 1)xi
rIi−.
A2,i = γλir + λi − µ r + λi
xi

xi =

(9)
(10)
(11)

The startup’s optimal sellout policy is to accept an acquisition only if the oﬀer arrives at time
t satisfying X(t) ≥ xi . As in the standard real options argument, if sunk cost I is present, the firm
waits for a suﬃciently good economic environment for sellout. By (9), I can show that limλi →0 xi =PVPV
xN
and limλi →∞ xi = xNβ/(β − 1). In fact, the sellout threshold xi monotonically increasesiiin the arrival rate λi . Intuitively, this means that in case of lower acquisition oﬀer frequency, thestartup accepts an acquisition oﬀer more eagerly in order to alleviate the illiquidity eﬀect. However,as the numerical examples in Section 4 show, this earnest sellout policy does not fully oﬀset theilliquidity eﬀects; in fact, the sellout timing is delayed with lower λi . Note that the sellout time is
not the same as, but comes after, the first hitting time to the sellout threshold xi , because the firm
has to wait for a Poisson jump.10 Thus, a lower λi decreases xi but delays the sellout time.
Next, I explain the firm value Vi (x) in Proposition 1. From the problem definition, Vi (x)monotonically increases in λi . By (8), I can show that limλi →0 Vi (x) = max{π(x), ai π(x) − I}(i.e., the NPV) and that limλi →∞ Vi (x) agrees with the real options value in the liquid model.
That is, a lower acquisition oﬀer frequency decreases the firm value by the embedded sellout option
value. This is consistent with the empirical observations of Song and Walkling (2000) and Cornett,
Tanyeri, and Tehranian (2011). Next, I explain expression (8) in more detail. For x < xi , π(x)is the NPV of perpetual cash flows, and A1,i xβ represents the value of the sellout option. In thisregion, where the firm does not accept an acquirer, Vi (x) > ai π(x) − I holds. For x ≥ xi , the first
three terms represent the expected payoﬀ of sellout as soon as an acquirer arrives, and the lastterm A2,i xγλi represents the value of the option to postpone sellout when an acquirer arrives forX(t) < xi . In this region, where the firm accepts an acquirer, Vi (x) ≤ ai π(x) − I holds.
Proposition 1 shows how the illiquidity of sellout opportunities aﬀects the startup’s sellout
policy and option value. However, owing to the homogeneity of acquirers, this proposition entails
no implication about the startup’s pricing policy (i.e., which type acquires the startup and how
much the acquirer pays). In the next section, I incorporate acquirer heterogeneity into the modeland explore the startup’s timing and pricing decisions simultaneously.

3.2

Symmetric information caseThis subsection assumes that λi > 0 (i = H, L), and that the acquirer types are observable to thestartup. I denote the firm value when an acquirer is absent as V (x). When an i-type acquirer
10

Mathematically, the probability of a Poisson jump occurring at the first hitting time is zero.

8

arrives, the firm value becomes max{ai π(X(t)) − I, V (X(t))}, because the startup optimizes the
choice between accepting and not accepting the i-type acquirer. As in Dupuis and Wang (2002), I
can show that the optimal policy is a threshold policy. In other words, the acceptance region for
an i-type acquisition becomes {x ≥ x∗i }, where x∗H < x∗L ≤ ∞, and V (x) satisfies ODEs:µxV ′ (x) + 0.5σ 2 x2 V ′′ (x) + x = rV (x)(0 < x < x∗H ),(12)µxV ′ (x) + 0.5σ 2 x2 V ′′ (x) + λH (aH π(x) − I − V (x)) + x = rV (x)µxV ′ (x) + 0.5σ 2 x2 V ′′ (x) + λ̄(āπ(x) − I − V (x)) + x = rV (x)(x∗H < x < x∗L ),(x > x∗L ),

(13)
(14)

where I define λ̄ = λH +λL and ā = (λH aH +λL aL )/λ̄. In (13), λH (aH π(x)−I −V (x)) corresponds
to the fact that V (x) becomes aH π(x) − I with probability λH dt in infinitesimal time interval dt,
and in (14), λ̄(āπ(x) − I − V (x)) corresponds to the fact that V (x) becomes aH π(x) − I withprobability λH dt and aL π(x) − I with probability λL dt in infinitesimal time interval dt. As in
Vi (x) in Section 3.1, V (x) is continuously diﬀerentiable at x∗i by Theorem 4.4.9 in Karatzas andShreve (1998), and V (x) satisfies boundary conditions
V (x∗i ) = ai π(x∗i ) − I(i = H, L),(15)

lim V (x) = 0,(16)

lim V (x)/x < ∞.

(17)

x→0

x→∞

Condition (15) corresponds to condition (3) in Section 3.1, meaning that the startup is indiﬀerent
to whether to accept or not accept an i-type acquirer at X(t) = x∗i (i.e., the optimality of x∗i ).
Conditions (16) and (17) correspond to conditions (4) and (5) in Section 3.1.
If
VH (x) ≥ aL π(x) − I(x ≥ 0),(18)which is equivalent to condition (19) in the following proposition, the startup takes the high-pricestrategy, that is, x∗L = ∞, where the startup always rejects a low type. This is because the firmcannot increase the firm value beyond VH (X(t)) by gaining payoﬀ aL π(X(t)) − I for any X(t). In
this case, the firm value V (x) and sellout threshold x∗H agree with VH (x) and xH in Proposition
1. Otherwise, the startup would take the flexible strategy, that is, x∗L < ∞, where the startup
accepts even a low type for X(t) ≥ x∗L . This is because the firm can increase the firm value
beyond VH (X(t)) by accepting a low type. In this case, by solving ODEs (12)–(14) with boundary
conditions (15)–(17), I have the following proposition. For proof of the proposition, see Appendix
B.
Proposition 2 If conditionaL − 1 ≤λH (aH − 1)r + λH − µ(19)holds, the startup takes the high-price strategy, that is, x∗L = ∞. The firm value V (x) and sellout
threshold x∗H agree with VH (x) and xH in Proposition 1.

9

If condition (19) does not hold, the startup takes the flexible strategy, that is, x∗L < ∞. The
firm value V (x) is given by
π(x) + B1 xβλH (aH − 1)π(x)λH Iπ(x) +−+ B2 xβλH + B3 xγλH
V (x) =
r
+
λ
−
µ
r
+ λH
H



λ̄(ā − 1)π(x)
λ̄I


−
+ B4 xγλ̄ π(x) +r + λ̄ − µr + λ̄

(x < x∗H ),(x ∈ [x∗H , x∗L )),(20)(x ≥ x∗L ),where coeﬃcients Bi (i = 1, 2, 3, 4) and sellout thresholds x∗i (i = H, L) are as defined in Appendix
B.
In case of heterogeneity of acquirers, the startup faces the dilemma of choosing sellout at ahigh price as well as with good timing eﬃciency. By Proposition 2, the startup’s policy can beclassified into two cases, the high-price strategy and flexible strategy based on the tradeoﬀ between
sellout pricing and timing eﬃciency. In the high-price strategy, the startup forgoes low types andwaits for a high type even in case of very good economic environment. In this sense, the highprice strategy pursues pricing eﬃciency rather than timing eﬃciency. In contrast, in the flexible
strategy, the startup accepts even a low-price acquisition in a very good economic environment (i.e.,X(t) ≥ x∗L ). That is, the flexible strategy purses timing eﬃciency rather than pricing eﬃciency.
Note that Proposition 2 shows the explicit condition (19) under which the startup takes thehigh-price strategy. Condition (19) can be rewritten asλ̄(ā − 1)
λH (aH − 1)≤,r + λH − µr + λ̄ − µ(21)where the left- and right-hand sides of (21) correspond to the second terms of equation (20) forx ≥ x∗L and equation (8) for x ≥ xH , respectively. Condition (21) means that the discounted
present value of aH − 1 at a high type’s first arrival time is not lower than the discounted present
value of ā − 1 at either type’s first arrival time. From (19), higher aH and λH as well as lower
aL and r − µ are likely to lead to the high-price strategy. The reason is as follows. Higher aH
and lower aL increase the advantage of high-price sellout over low-price sellout. Lower r − µ also
increases the advantage of high-price sellout, because r − µ is the denominator of π(x). Higher λH
decreases the timing ineﬃciency of waiting for only a high type. Thus, with higher aH and λH as
well as lower aL and r − µ, the startup purses the pricing eﬃciency stemming from the high-pricestrategy.
Next, I discuss the flexible strategy and its value, because the high-price strategy has alreadybeen explained after Proposition 1. In the flexible strategy case, the startup accepts only a hightype in an intermediate economic environment, that is, X(t) ∈ [x∗H , x∗L ). In this region, thestartup pursues the pricing eﬃciency stemming from high-price sellout. However, if X(t) rises
beyond x∗L , the startup accepts even a low type because the timing eﬃciency from accepting either
type dominates the pricing eﬃciency. In this region, the sellout timing as well as sellout price isunpredictable because it depends on the acquirer type. This result is consistent with the empiricalevidence of high uncertainty in M&A deals (e.g., Cornett, Tanyeri, and Tehranian (2011) and Wang(2018)).

10I cannot derive closed-form expressions of x∗i (i = H, L), but by the monotonic increase of V (x)with respect to λi (i = H, L) and x∗i = inf{x ≥ 0 | V (x) ≤ ai π(x) − I}, I can readily show thatx∗i (i = H, L) monotonically increases in λi (i = H, L). This means that with lower frequency ofacquisition oﬀers, the startup is more eager to accept an acquisition and alleviate the illiquidity
eﬀect. However, as shown in the numerical examples in Section 4, this earnest sellout policy does
not fully oﬀset the illiquidity eﬀects; in fact, the sellout timing is delayed with lower λi . These
results are in line with Proposition 1.
Now, I explain the firm value (20) in Proposition 2. For x < x∗H , term B1 xβ stands for thevalue of sellout option. In this case, where the firm rejects any acquirer, V (x) > aH π(x) − I holds.
For x ∈ [x∗H , x∗L ), the first three terms represent the expected payoﬀ of sellout to the high type that
arrives first. In this case, where the firm accepts only a high type, aL π(x) − I < V (x) ≤ aH π(x) − I
holds. The terms B2 xβλH + B3 xγλH represent the option value to change the policy when X(t)passes either x∗H or x∗L before the arrival of a high type. For x ≥ x∗L , the first three terms represent
the expected payoﬀ of sellout to the acquirer that arrives first. In this case, where the firm accepts
either type, V (x) ≤ aL π(x) − I holds. The last term B4 xγλH represents the option value to change
the policy when X(t) passes x∗L before the acquirer’s arrival.
From a technical viewpoint, Proposition 2 extends the results of Dupuis and Wang (2002)(or equivalently Proposition 1) to the case with heterogeneity in option payoﬀs. Proposition 2 also
clarifies the relationship between heterogeneous and homogeneous problems by deriving the explicit
condition (19) under which the heterogeneous problem reduces to the homogeneous problem. The
solution in the flexible strategy case is novel, but is similar to that in Morellec, Valta, and Zhdanov(2015) and Nishihara and Shibata (2021). In the investment and financing model of Morellec,
Valta, and Zhdanov (2015), for intermediate levels of the state variable, a firm invests only whena private financing opportunity is available, but invests with public debt financing when the statevariable exceeds a certain threshold. In the bankruptcy model of Nishihara and Shibata (2021), forintermediate levels of the state variable, a firm exits only when sellout is feasible, but bankrupts
when the state variable decreases to a certain threshold. Unlike this paper, previous studies consider
two options, one (i.e., private debt finance, sellout) feasible only at Poisson jump times, and theother (i.e., public debt finance, default) feasible at any time.

3.3

Asymmetric information caseThis subsection assumes that λi > 0 (i = H, L) and the acquirer types are unobservable to thestartup. I denote the firm value when an acquirer is absent as U (x). When an acquirer arrives,
the startup makes a take-it-or-leave-it price oﬀer. The startup posts a price from among aL π(x) −Ia , aH π(x) − Ia , and ∞ (i.e., rejection), because it cannot increase the firm value by any otherprice. The acquirer then purchases the firm if the posted price is not higher than its evaluation. As
in Sections 3.1 and 3.2, the optimal policy becomes a threshold policy. In other words, the i-type

11

∗∗
∗∗
pricing region becomes {x ≥ x∗∗i }, where xH < xL ≤ ∞, and U (x) satisfies ODEs:µxU ′ (x) + 0.5σ 2 x2 U ′′ (x) + x = rU (x)(0 < x < x∗∗
H ),

(22)

µxU ′ (x) + 0.5σ 2 x2 U ′′ (x) + λH (aH π(x) − I − U (x)) + x = rU (x)µxU ′ (x) + 0.5σ 2 x2 U ′′ (x) + λ̄(aL π(x) − I − U (x)) + x = rU (x)∗∗(x∗∗
H < x < xL ),(x > x∗∗
L ).

(23)
(24)

ODEs (22) and (23) are the same as ODEs (12) and (13) in Section 3.2, respectively, but the third
term in ODE (24) is diﬀerent from that in ODE (14). Indeed, λ̄(aL π(x) − I − U (x)) indicates that
U (x) becomes aL π(x) − I with probability λ̄dt in infinitesimal time interval dt. In this region,
unlike in the symmetric information case, the sellout price is low even for a high type because thestartup cannot determine the price contingent on the acquirer type.
As in Sections 3.1 and 3.2, U (x) is continuously diﬀerentiable at x∗∗
i , and U (x) satisfies boundary
conditions∗∗
U (x∗∗
H ) = aH π(xH ) − I,

(25)

∗∗
U (x∗∗
L ) = (aL − ∆aλH /λL )π(xL ) − I,(26)

lim U (x) = 0(27)

lim U (x)/x < ∞.

(28)

x→0

x→∞

Boundary conditions (25), (27), and (28) are the same as (3), (4), and (5) in Section 3.2, respectively,
but (26) is diﬀerent from (3). The term ∆aλH /λL π(x∗∗L ) in (26) stands for the opportunity cost of
the low-price sale, that is, the loss from sellout to a high type at a low price. Here, condition (26)is derived as follows. Suppose that an acquirer arrives at the threshold x∗∗
L . By setting a low price,
the firm value becomes aL π(x∗∗L ) − I because the acquirer would purchase the startup. By setting a∗∗high price, the firm value becomes (aH π(x∗∗L ) − I)λH /λ̄ + U (xL )λL /λ̄ because the acquirer wouldpurchase the startup if and only if it is a high type. Note that the acquirer is a high type withprobability λH /λ̄. By solving equation∗∗∗∗aL π(x∗∗L ) − I = (aH π(xL ) − I)λH /λ̄ + U (xL )λL /λ̄,I obtain the boundary condition (26), which ensures that the startup is indiﬀerent to whether to∗
set a high or low price at X(t) = x∗∗
L (i.e., the optimality of xL ).

If(aH π(x) − I)λH /λ̄ + VH (x)λL /λ̄ ≥ aL π(x) − I(x ≥ 0)(29)

holds, which is equivalent to condition (31) in the following proposition, the startup takes thehigh-price strategy, that is, x∗∗
L = ∞, where the startup posts only the high price. This is because
the expected payoﬀ by the high-price strategy (i.e., the left-hand side of (29)) dominates the low
payoﬀ (i.e., the right-hand side of (29)) for any X(t). In this case, the firm value U (x) and sellout
threshold x∗∗
H agree with VH (x) and xH in Proposition 1. Condition (29) can be rewritten as
VH (x) ≥ aL π(x) − I − ∆aπ(x)λH /λL

12(x ≥ 0).

(30)

Compared to (18), (30) has an extra term −∆aπ(x)λH /λL , which is the opportunity cost of the
low-price sale, that is, the loss from sellout to a high type at a low price. If (29) (or equivalently(30) or (31)) does not hold, the startup takes the flexible strategy, that is, x∗∗
L < ∞, where thestartup allows the low-price sale for X(t) ≥ x∗∗
L . In fact, the firm can increase the firm value
beyond VH (X(t)) by setting the low price. In this case, by solving ODEs (22)–(24) with boundary
conditions (25)–(28), I have the following proposition. For proof of the proposition, see Appendix
C.
Proposition 3 If conditionaL − 1 −∆aλH
λH (aH − 1)≤λL
r + λH − µ(31)holds, the firm takes the high-price strategy, that is, x∗∗
L = ∞. The firm value U (x) and sellout
threshold x∗∗
H agree with VH (x) and xH in Proposition 1.
If condition (31) does not hold, the firm takes the flexible strategy, that is, x∗∗
L < ∞. The firm
value U (x) is given by π(x) + B̃1 xβ
λH (aH − 1)π(x)λH Iπ(x) +−+ B̃2 xβλH + B̃3 xγλH
U (x) =
r
+
λ
−
µ
r
+ λH
H



λ̄I
λ̄(aL − 1)π(x)−+ B̃4 xγλ̄ π(x) +r + λ̄ − µr + λ̄

(x < x∗∗H ),∗∗(x ∈ [x∗∗H , xL )),(32)(x ≥ x∗∗L ),where the coeﬃcients B̃i (i = 1, 2, 3, 4) and sellout thresholds x∗∗(i = H, L) are as defined iniAppendix C.
As in Proposition 1 in Section 3.2, Proposition 2 shows that the startup chooses either thehigh-price strategy, abandoning the low-price sellout to pursue pricing eﬃciency, or the flexible
strategy, adopting the low-price sellout to pursue timing eﬃciency. The choice is determined bywhether condition (31) holds or not. Compared to condition (19), the left-hand side of (31) has an
extra term −∆aλH /λL . As with the diﬀerence between (18) and (30), the extra term represents
the opportunity cost of the low-price sellout, that is, the loss from sellout to a high type at a low
price. In other words, with asymmetric information, the pricing ineﬃciency stemming from theflexible strategy increases, and the startup is more likely to choose the high-price strategy.
Condition (31) can be rewritten as
λ̄(aL − 1)
λH (aH − 1)≤,r + λH − µr + λ̄ − µ(33)

where the left- and right-hand sides of (21) correspond to the second terms of equation (32) forx ≥ x∗∗
L and equation (8) for x ≥ xH , respectively. As in the symmetric information case, withhigher aH and λH as well as lower aL and r − µ, the startup is more likely to take the high-pricestrategy. Note that from condition (31), the firm tends to take the high-price strategy with higher∆a and lower λL . This is explained by the opportunity cost of the low-price sellout. The pricing
ineﬃciency increases with higher ∆a, and the probability of the acquirer type being high (i.e.,λH /λ̄) increases with lower λL . Hence, with higher ∆a and lower λL , the startup tends to take thehigh-price strategy in order to remove the pricing ineﬃciency.

13

In the region aL − 1 ≤ λH (aH − 1)/(r + λH − µ), the startup takes the high-price strategy, thatis, the strategy with only high type (i.e., Proposition 1), regardless of the information structure.
Here, asymmetric information does not aﬀect the sellout policy and option value, i.e., U (x) =V (x) = VH (x). In the region aL − 1 − ∆aλH /λL ≤ λH (aH − 1)/(r + λH − µ) < aL − 1, the startup
takes the high-price strategy under asymmetric information and flexible strategy under symmetric
information. Thus, asymmetric information changes the strategy and reduces the firm value, i.e.,
U (x) = VH (x) < V (x). In the region λH (aH − 1)/(r + λH − µ) < aL − 1 − ∆aλH /λL , the startup
takes the flexible strategy irrespective of the information structure. However, the sellout thresholds∗x∗∗i (i = H, L) are diﬀerent from xi (i = H, L), and the firm value decreases with asymmetric

information, i.e., U (x) < V (x).
I cannot derive closed-form expressions of x∗∗i (i = H, L), but by the monotonic increase of∗∗
U (x) with respect to λi (i = H, L), x∗∗
H = inf{x ≥ 0 | U (x) = aH π(x) − I}, and xL = inf{x ≥ 0 |U (x) = (aL − ∆aλH /λL )π(x) − I}, I can show that x∗∗
H monotonically increases in λi (i = H, L)∗∗∗
and x∗∗L monotonically increases in λH . By U (x) < V (x), I can also show that xH < xH . Although∗∗∗
I cannot analytically prove this, I find that x∗∗
L monotonically decreases in λL , and xL > xL holds

in the numerical examples in Section 4.
From these results, in case of lower frequency of high type arrivals, the startup accepts an
acquisition more eagerly to alleviate the illiquidity eﬀect. These results agree with Propositions 1
and 2. More notably, in case of lower frequency of low type arrivals, the startup accepts a highprice acquisition more eagerly and a low-price acquisition more restrictively. This is because theopportunity cost of a low-price sellout increases with a lower λL . Then, despite decreased liquidity,
the startup restricts the low-price sellout more severely. This is a key diﬀerence from the symmetric
information case. In case of asymmetric information, the startup extends the gap between the high∗∗
price sellout threshold x∗∗
H and low-price sellout threshold xL to reduce the high type’s information

rent. This result agrees with the previous literature on intertemporal price discrimination (e.g.,
Stokey (1979), Landsberger and Meilijson (1985), Nishihara and Shibata (2019)). In fact, Stokey(1979) and Landsberger and Meilijson (1985) show that a seller can delay sales at low prices todiscriminate between buyer types, while Nishihara and Shibata (2019) show that a firm can delay
sellout or the default timing for a low type to decrease the acquirer’s information rent.
Proposition 3 technically contributes to the literature on the optimal stopping problem constrained to Poisson jump times. Indeed, previous papers (e.g., Dupuis and Wang (2002), Morellec,
Valta, and Zhdanov (2015), Nishihara and Shibata (2021)) lack in heterogeneity and hence do notconsider any problem under asymmetric information. To my knowledge, Proposition 3 first solves
the problem involving heterogeneity of option payoﬀs and asymmetric information.
Although the acquirer’s surplus is zero under symmetric information, a high type can obtain
information rent by acquiring the firm at a low price under asymmetric information. The following
corollary derives the expected acquirer payoﬀ at time 0 (denoted by W (x)). For proof of thecorollary, see Appendix D.

14

Corollary 1 If condition (31) is satisfied, W (x) = 0. If condition (31) is not satisfied, W (x) is
given by
D 1 xβ
D2 xβλH + D3 xγλH
W (x) =

λH ∆aπ(x)+ D4 xγλ̄r + λ̄ − µ(x < x∗∗H ),∗∗(x ∈ [x∗∗H , xL )),(34)(x ≥ x∗∗L ),where the sellout thresholds x∗∗i (i = H, L) and coeﬃcients Di (i = 1, 2, 3, 4) are defined as inAppendices C and D.
The positive acquirer’s information rent is consistent with empirical evidence. In fact, many studies
such as Bargeron, Schlingemann, Stulz, and Zutter (2008), Battigalli, Chiarella, Gatti, and Orlando(2017), and Golubov and Xiong (2020) show that private and less transparent firms pay lower
acquisition premiums. I define the social loss due to asymmetric information as L(x) = V (x) −U (x) − W (x). In the next section, I examine L(x) to study the welfare eﬀects of asymmetric
information.
Throughout this study, I assume that the acquirer pays cash to acquire the startup. Although
the results under symmetric information (i.e., Propositions 1 and 2) hold true regardless of theacquirer’s payment method, the results under asymmetric information (i.e., Proposition 3 andCorollary 1) depend on the payment method. In fact, using of stock and payment contingent
on the post-merger performance (e.g., Lukas, Reuer, and Welling (2012)) decreases the acquirer’s
information rent. The startup would then prefer such a contract, whereas the acquirer would prefer
cash payment to maximize its information rent. The conflict over payment method is beyond thescope of this study, but would be an interesting topic for a future research.

4

Numerical analysis

In this section, I carry out numerical analysis, including comparative statics of the arrival rates
λi (i = H, L), cash flow growth rate µ, and volatility σ. I set the baseline parameter values as in
Table 1. The values of r, µ, and σ are typical to the real options literature (e.g., Dixit and Pindyck(1994)). In particular, I set µ and σ so as to satisfy µ − 0.5σ 2 > 0, meaning that X(t) has a growing
trend, and focus on the startup in the growth stage.
Figure 1 shows how the startup’s sellout strategy changes with the expected waiting time
1/λi (i = H, L). With suﬃciently frequent high type arrivals (i.e., 1/λH ≤ 5), the startup takes
the high-price strategy regardless of the information structure. In the region between the black andblue dashed lines, the firm takes the high-price strategy only with asymmetric information, whereas,in the right-hand side region, the firm takes the flexible strategy regardless of the information
structure. I set the baseline parameters λH = 0.125 and λL = 0.375 (see the baseline point
in Figure 1) so as to satisfy conditions (19) and (31), and examine mainly the flexible strategy
case and the eﬀects of asymmetric information. These values mean that on average, an acquirer

15

approaches the startup every two years (i.e., 1/λ̄ = 2), and low types arrive three times more
frequently than high types (i.e., λL /λH = 3).
For the baseline parameter values, I have x∗H = 1.504, x∗L = 5.472, and V (x) = 22.382 under∗∗symmetric information and x∗∗
H = 1.5, xL = 24.606, and U (x) = 22.374 under asymmetric infor-

mation. As explained after Proposition 3, with asymmetric information, the startup increases thegap between the high- and low-price sellout thresholds to decrease the acquirer’s information rent.
∗
In particular, the firm takes x∗∗
L much higher than xL . The diﬀerence between V (x) and U (x) may

look very small because the NPV of cash flows with no sellout, that is, π(x) = 20, accounts formost of V (x) and U (x).

4.1

Impacts of high type arrival rate

Figure 2 plots the thresholds x∗i , x∗∗i , firm values V (x), U (x), and social loss L(x) with varying
levels of expected waiting time for the high type, 1/λH . The other parameter values are set asin Table 1. In addition, the figure plots the probability of high-price sellout (denoted by PH (x)and QH (x) under symmetric and asymmetric information, respectively) and state price of sellout(denoted by S(x) and T (x) under symmetric and asymmetric information, respectively). The state
prices, defined by S(x) = E[e−rτ ] and T (x) = E[e−rτ̃ ], where τ and τ̃ denote the sellout time undersymmetric and asymmetric information, respectively, measure the time until sellout.
Condition (19) in Proposition 2 holds for 1/λH ≤ 5, whereas condition (31) in Proposition 3
holds for 1/λH ≤ 7.6. In these regions, the startup takes the high-price strategy, where x∗L and x∗∗Lare infinity, and PH (x) and QH (x) are equal to 1. As explained after Propositions 2 and 3, in the
top panels, x∗i and x∗∗i (i = H, L) decrease in 1/λH . In particular, I find the eﬀects of 1/λH onx∗L and x∗∗
L very large, meaning that the firm’s response to a low-price acquisition depends greatly
on the frequency of high type arrivals. These results show that with lower frequency of high typearrivals, the startup accepts an acquisition more eagerly (especially of a low type) to alleviate theilliquidity eﬀect.
The center-right panel shows that PH (x) and QH (x) decrease in 1/λH ; this implies that lowprice sellout is more likely to occur with lower frequency of high type arrivals. Note that the large
decrease in x∗L and x∗∗
L with higher 1/λH also contributes to this result. The bottom-left panel
shows that S(x) and T (x) decrease in 1/λH , meaning that lower frequency of high type arrivals
lengthens the expected time until sellout. To summarize, with lower frequency of high type arrivals,
the firm accepts an acquirer more eagerly, but this earnest sellout policy does not fully oﬀset thedecreased frequency eﬀect. In fact, lower frequency of high type arrivals delays the sellout timeand reduces the high-price sellout probability and firm value.
Next, I examine the diﬀerences between the results under symmetric and asymmetric informa∗∗∗∗∗tion. In the top panels of Figure 2, as explained after Proposition 3, x∗∗
H ≤ xH and xL > xH

hold. The center-right panel shows that QH (x) is higher than PH (x), while the bottom-left panel
shows that T (x) is lower than S(x). These results are interpreted as follows. With asymmetric
information, the startup accepts a high-price sellout more eagerly and a low-price sellout more16

restrictively to reduce the acquirer’s information rent. Thus, the startup can keep the sellout pricehigh, but this delays the sellout timing.
The welfare eﬀects of asymmetric information are not monotonic with respect to 1/λH . In
fact, the bottom-right panel shows that social loss L(x) has an inverted U-shape in 1/λH . The
reason is that asymmetric information matters most for intermediate levels of 1/λH . For 1/λH ≤ 5,∗
there is no diﬀerence between the symmetric and asymmetric information cases, i.e., x∗∗
H = xH =∗xH , x∗∗
L = xL = ∞, U (x) = V (x) = VH (x), PH (x) = QH (x) = 1, and L(x) = 0. For 1/λH → ∞,the firm does not meet a high type, and therefore asymmetric information does not matter, i.e.,∗x∗∗
L = xL = xL , U (x) = V (x) = VL (x), PH (x) = QH (x) = 0, and L(x) = 0. For intermediatelevels of 1/λH , especially in the region of 5 < 1/λH ≤ 7.6, where the startup takes the high-pricestrategy only in the presence of asymmetric information, the eﬀects of asymmetric information
become relatively large, leading to high L(x).

4.2

Impacts of low type arrival rate

Figure 3 plots x∗i , x∗∗i , V (x), U (x), PH (x), QH (x), S(x), T (x), and L(x) with varying levels of expected waiting time for a low type, 1/λL . The other parameter values are set as in Table 1. In all
the depicted regions, the startup takes the flexible strategy under symmetric information, and theflexible strategy for 1/λL ≤ 3.3 and high-price strategy for 1/λL > 3.3 under asymmetric information. As explained after Proposition 3, the opportunity cost of low-price sellout increases withhigher 1/λL , and the startup would then be more likely to choose the high-price strategy.
As explained after Proposition 2, with symmetric information, x∗i (i = H, L) decrease in 1/λLin the top panels and S(x) decreases in 1/λL in the bottom-left panel. This can be interpreted
as follows. With lower frequency of low type arrivals, the startup would accept an acquisition
more eagerly to alleviate the decreased frequency eﬀect. However, this earnest policy will not fully
oﬀset the decreased frequency eﬀect and delay the sellout. These impacts of 1/λL under symmetric
information are similar to but weaker than those of 1/λH in Section 4.1.
∗∗
More interestingly, a higher 1/λL decreases x∗∗
H but increases xL under asymmetric information.

The eﬀect of 1/λL on x∗∗
L is quite large and diﬀerent from that of 1/λH or the symmetric information
case. As explained after Proposition 3, this result stems from the opportunity cost of low-pricesellout. In fact, the probability of high acquirer type increases with higher 1/λL . Then, with higher
1/λL , the startup would restrict low-price sellout more severely to mitigate the risk of sellout toa high type at a low price, although it would accept a high-price sellout more eagerly. By thissellout policy under asymmetric information, the high-price sellout probability QH (x) increases
more sharply than PH (x) in the center-right panel of Figure 3, but the state price T (x) decreases
more sharply than S(x) in the bottom-left panel. In other words, the startup optimizes the sellout
policy with a tradeoﬀ between sellout pricing and timing eﬃciency.
As in 1/λH in Section 4.1, the welfare eﬀects of asymmetric information are not monotonic withrespect to 1/λL . For 1/λL → 0, the firm meets a low type at a low-price sellout threshold, and∗hence asymmetric information does not matter, i.e., x∗∗
L = xL , U (x) = V (x),, and L(x) = 0. For17

1/λL → ∞, a low type never arrives, and therefore asymmetric information does not matter, i.e.,∗x∗∗
H = xH = xH , U (x) = V (x) = VH (x), PH (x) = QH (x) = 1, and L(x) = 0. For intermediate levelsof 1/λL , the eﬀects of asymmetric information become relatively large. Note that V (x) − U (x) andL(x) are highest especially for 1/λL ≈ 3.3, where the firm takes the high-price strategy only withasymmetric information.

4.3

Impacts of cash flow growth rate

Figure 4 plots x∗i , x∗∗i , V (x), U (x), PH (x), QH (x), S(x), T (x), and L(x) with varying cash flow growthrate levels, µ. The other parameter values are set as in Table 1. The startup takes the flexible
strategy for µ < 0.033 and high-price strategy for µ ≥ 0.033 under symmetric information, and theflexible strategy for µ < 0.049 and high-price strategy for µ ≥ 0.049 under asymmetric information.
As conditions (19) and (31) explain, a higher µ, which magnifies the gap between high and low
valuations, intensifies the pricing ineﬃciency stemming from the flexible strategy, and hence thestartup is more likely to take the high-price strategy.
11
In Figure 4, x∗H and x∗∗
H decrease in µ, whereas V (x), U (x), S(x), and T (x) increase in µ.

These comparative static results are consistent with the standard results (e.g., Dixit and Pindyck(1994)). In fact, a higher µ is known to accelerate the option exercise timing and increases the
project value. More notably, x∗L and x∗∗
L increase in µ in the top-right panel, in contrast to thestandard result. A higher µ expands the gap between the high and low sellout prices, and therefore
the startup accepts a high-price sellout more eagerly and a low-price sellout more restrictively.
Thus, the high-price sellout probabilities PH (x) and QH (x) increase in µ. These results lead tothe empirical prediction that startups with higher growth rates are more likely to be acquired byhigher-eﬃciency firms. To my knowledge, this prediction has not been tested empirically.
The welfare eﬀects of asymmetric information are not monotonic with respect to µ.

Forµ ≥ 0.049, where the startup takes the same high-price strategy regardless of information structures, asymmetric information makes no diﬀerence to welfare. As in the previous subsections, forintermediate levels of µ, especially µ ≈ 0.033, where the firm changes the flexible strategy to thehigh-price strategy under asymmetric information, L(x) becomes relatively high.

4.4

Impacts of cash flow volatility

Figure 5 plots x∗i , x∗∗i , V (x), U (x), PH (x), QH (x), S(x), T (x), and L(x) with varying levels of cash
flow volatility, σ. The other parameter values are set as in Table 1. Neither condition (19) nor
condition (31) depends on σ, and the startup therefore takes the flexible strategy in all areas.
In Figure 5, x∗i , x∗∗i , V (x), and U (x) increase in σ, whereas S(x) and T (x) decrease in σ. These
comparative static results are consistent with the standard result (e.g., Dixit and Pindyck (1994))

11

In the figure, x∗H , V (x), and S(x) almost align with x∗H , U (x), and T (x), respectively, because the impacts of µ onthese values are much stronger than those of asymmetric information.

18

that a higher σ delays the option exercise timing and increases the value of waiting.12 More notably,
the high-price sellout probabilities PH (x) and QH (x) decrease in σ, in the center-right panel. Thisis because a higher σ, which makes the dynamics of X(t) more volatile, increases the probability∗∗
that X(t) passes through interval [x∗H , x∗L ) (or [x∗∗H , xL )) without meeting a high type. In otherwords, higher volatility can rapidly improve the economic environment such that the startup cangain suﬃciently high proceeds even from a low-price sellout. These results lead to the empiricalprediction that startups with higher volatility are more likely to be acquired by lower-eﬃciency
firms. To my knowledge, this prediction has not been tested empirically.
Figure 5 shows that the impacts of asymmetric information increase monotonically in σ. In fact,∗∗∗
the diﬀerences, x∗H − x∗∗
H , xL − xL , V (x) − U (x), PH (x) − QH (x), and S(x) − T (x), as well as social

loss L(x), increase in σ. The reason seems to be the same as discussed above. That is, with higher σ,which has the potential to increase X(t) rapidly, a low-price sellout matters more. This increases theineﬃciency due to asymmetric information because a low-price sellout with asymmetric information
leads to loss from sellout to a high type at a low price. To alleviate the increased ineﬃciency,∗∗∗
the startup adjusts the sellout policy more, leading to x∗H − x∗∗
H , xL − xL , PH (x) − QH (x), and

S(x)−T (x) increasing in σ. However, this adjustment does not fully oﬀset the increased ineﬃciency,
and V (x) − U (x) and L(x) therefore increase in σ.

4.5

Stock price reaction at sellout

Figure 6 shows the expected jump in startup value at the sellout time under symmetric and asymmetric information. With symmetric information, acquisition changes the firm value from V (X(t))
to aH π(X(t)) − I for X(t) ∈ [x∗H , x∗L ), and to ai π(X(t)) − I for X(t) ≥ x∗L . Then, the expectedjump size becomes (aH π(X(t)) − I − V (X(t)))/V (X(t)) for X(t) ∈ [x∗H , x∗L ), and (āπ(X(t)) − I −V (X(t)))/V (X(t)) for X(t) ∈ [x∗H , x∗L ). Under asymmetric information, the jump size becomes∗∗(aH π(X(t)) − I − U (X(t)))/U (X(t)) for X(t) ∈ [x∗∗H , xL ) and (aL π(X(t)) − I − U (X(t)))/U (X(t))

for X(t) ∈ [x∗H , x∗L ). The jump size can be interpreted as the target stock price reaction or acquisition premium.
From Figure 6, the jump size for the flexible strategy is not monotonic with respect to X(t).
In fact, the jump size at acquisition with symmetric information increases monotonically in X(t)for X(t) ∈ [x∗H , x∗L ), and X(t) > x∗L , but falls at X(t) = x∗L . This same result can be found underasymmetric information. This can be explained as follows. At the threshold x∗L (or x∗∗
L ), the startupchanges its sellout strategies and accepts a low-price acquisition. By allowing a low-price acquisition,
the sellout probability increases, whereas the sellout price decreases. Then, despite better economic
environment, the jump size decreases. In particular, under asymmetric information, the jump size
reduces more sharply at the threshold x∗∗
L . This result leads to the empirical prediction thatasymmetric information intensifies stock price reactions at acquisition.
Figure 6 also shows that a higher 1/λH leads to larger jump sizes as long as the sellout region
12

In contrast, Sarkar (2021) shows that the relationship between uncertainty and investment is not monotonic in the

optimal investment timing and size problem.

19

does not change. This is because the firm values V (X(t)) and U (X(t)) decrease with higher 1/λH .
In other words, with higher 1/λH , the firm value includes less sellout option values, and the sellout
therefore brings a larger shock to the firm value. This result is consistent with the empiricalfindings of Cornett, Tanyeri, and Tehranian (2011) that a lower merger anticipation increases thetarget stock price reaction at merger announcement. More notably, the jump size is not monotonic
when the sellout region changes with 1/λH . For instance, see the reactions for X(t) = 10. Undersymmetric information, the jump size increases with 1/λH = 8, 6, and 10. Under asymmetric
information, the jump size increases with 1/λH = 10, 8, and 6. These results show the ambiguous
relationship between merger anticipation and target stock price reaction. Indeed, a lower merger
anticipation does not necessarily increase the stock price reaction because it may also decrease theprice that the target firm aims for. I omit a figure with respect to 1/λL because the same results
hold true.
Most of the M&A real option models (e.g., Lambrecht (2004), Morellec and Zhdanov (2008),
and Lukas, Pereira, and Rodrigues (2019)) do not consider the friction in the M&A searching
and matching process. Hence, those models can predict the M&A timing, and therefore no jumpoccurs in firm value at the time of M&A. Hackbarth and Morellec (2008) consider the asymmetric
information between firm insiders and outside investors, and examine the jump in firm value, but
the target and acquirer managers in their models precisely predict the merger timing and price.
Nishihara and Shibata (2021) consider the illiquidity of sellout opportunities, but do not consider
the heterogeneity in sellout prices. This study can better account for the empirical findings of highuncertainty in M&A deals (e.g., Cornett, Tanyeri, and Tehranian (2011) and Wang (2018)), positive
target stock price reactions (e.g., Huang and Walkling (1987) and Boone and Mulherin (2007)),and lower acquisition premiums oﬀered by less transparent acquirers (e.g., Bargeron, Schlingemann,
Stulz, and Zutter (2008), Battigalli, Chiarella, Gatti, and Orlando (2017), and Golubov and Xiong(2020)).

5

Conclusion

This study develops a model in which a startup optimizes the sellout timing and price for highand low-type acquirers approaching randomly and sequentially. Under symmetric and asymmetric
information on acquirer type, I analytically derive the startup’s sellout policy and firm value. In
particular, the startup takes either the high-price strategy—accepting only a high type— or flexible
strategy—accepting even a low type in a very good economic environment—based on a tradeoﬀ
between sellout pricing and timing eﬃciency. The model analysis yields the following results.
The startup tends to take the high-price strategy when the high type’s arrival rate and cash flow
growth rate are high. With lower acquirers’ arrival rate, the startup accepts an acquisition moreeagerly, but this policy does not fully oﬀset the decrease in liquidity. In fact, lower arrival rates
delay the sellout timing and decrease the firm value. A higher growth rate increases the low-pricesellout threshold because it magnifies the gap between high and low valuations. Higher growth20

rate and lower volatility increase the high-price sellout probability and accelerate the sellout. Withasymmetric information, the high-type acquirer can collect information rent by purchasing thestartup at a low price. To reduce the acquirer’s information rent, the startup decreases the highprice sellout threshold and increases the low-price sellout threshold, especially for low arrival rates
of low types. This policy adjustment maintains the sellout price, but delays the sellout time, leading
to social loss. This social loss due to asymmetric information tends to be high when asymmetric
information changes the startup’ strategy from flexible to high-price strategy. The model can also
explain the positive target stock price reactions (or acquisition premiums) at acquisition. The jump
size can be non-monotonic with respect to either the state variable or acquirer arrival rate due tothe interaction between the sellout price and timing.

A

Proof of Proposition 1

The general solution to ODE (1) with (4) is expressed as Vi (x) = π(x) + A1,i xβ , where A1,i is an
unknown coeﬃcient. Then, by (5) and the continuity of Vi (x), I have the expression (10). The
general solution to ODE (2) with (5) is expressed as
Vi (x) = π(x) +λi (ai − 1)π(x)λi I−+ A2,i xγλi ,r + λi − µr + λiwhere A2,i is an unknown coeﬃcient. Then, by (3) and the continuity of Vi (x), I have the expression(11). By the continuous diﬀerentiability of Vi (x) at xi , I haveβA1 xβi =λi (ai − 1)π(xi )γλ+ γλi A2 xi i .
r + λi − µ(35)

By substituting (10) and (11) into (35), I haveβ((ai − 1)π(xi ) − I) =xi =λi (ai − 1)π(xi )+ γλi
r + λi − µ((ai − 1)xi
rI−
r + λi − µ r + λi)PV(r + λi − µ)((β − γλi )r + βλi )xN
i,(r + λi )((β − γλi )(r − µ) + (β − 1)λi )PVwhere xN
= (r − µ)I/(ai − 1).
i

BProof of Proposition 2First, assume that (19) holds. Now, I can show that VH (x) > aL π(x) − I (x ≥ 0) as follows. Forx < xH , I have
VH (x) > aH π(x) − I ≥ aL π(x) − I.

(36)

By A2,H > 0 in (8), I have, for x ≥ xH ,λH (aH − 1)π(x)λH I−r + λH − µr + λHλH I≥ aL π(x) −r + λH

VH (x) > π(x) +> aL π(x) − I,

21

(37)
(38)

where (37) follows from (19). By (36) and (38), I have VH (x) > aL π(x) − I for all x ≥ 0. I also
have V (x) > aL π(x) − I for all x ≥ 0 because of V (x) ≥ VH (x). Then, there is no acceptanceregion for a low type, and V (x) = VH (x) holds.
Next, assume that (19) does not hold. From (8), I have VH (x) < aL π(x) − I when x is
suﬃciently large. Hence, I have V (x) ̸= VH (x), meaning that the firm accepts a low type. Assume
that the value function V (x) exists. Then, I can straightforwardly show that an optimal policy
exists among threshold policies specified by thresholds x∗i (i = H, L), where x∗H < x∗H , and that
V (x) is an increasing, convex, and continuously diﬀerentiable function satisfying ODEs (12)–(14)with boundary conditions (15)–(17).
The solution to ODE (12) with boundary conditions (15) for i = H and (16) becomes (20) forx < x∗H . Similarly, the solution to ODE (13) with boundary condition (15) for i = H, L is (20) forx ∈ (x∗H , x∗L ), while the solution to ODE (14) with boundary conditions (15) for i = L and (17) is(20) for x > x∗L . Notations Bi (i = 1, . . . , 7) are defined by
B1 = (x∗H )−β ((aH − 1)π(x∗H ) − I),
B2 =(x∗H )γλH B6 − (x∗L )γλH B5,(x∗L )βλH (x∗H )γλH − (x∗H )βλH (x∗L )γλH

B3 =(x∗L )βλH B5 − (x∗H )βλH B6
,
βλH
∗
(xL )
(x∗H )γλH − (x∗H )βλH (x∗L )γλH

B4 = (x∗L )−γλ̄ B7 ,rI
(aH − 1)x∗H−,r + λH − µr + λHrIλH (aH − 1)π(x∗L )B6 = (aL − 1)π(x∗L ) −−,r + λH − µr + λHrIλ̄(ā − 1)π(x∗L )−.
B7 = (aL − 1)π(x∗L ) −r + λ̄ − µr + λ̄
B5 =Because of the continuous diﬀerentiability of V (x) at x∗i (i = H, L), (x∗H , x∗L ) becomes a solution13toλH (aH − 1)π(x∗H )+ βλH B2 (x∗H )βλH + γλH B3 (x∗H )γλH = β((aH − 1)π(x∗H ) − I),r + λH − µλ̄(ā − 1)π(x∗L )λH (aH − 1)π(x∗L )+ βλH B2 (x∗L )βλH + γλH B3 (x∗L )γλH =+ γλ̄ B7 .
r + λH − µ
r+λ−µ

C

(39)
(40)

Proof of Proposition 3

First, assume that (31) holds. In this case, I can show that(aH π(x) − I)λH /λ̄ + VH (x)λL /λ̄ ≥ aL π(x) − I (x ≥ 0)

13

(41)

I find a unique solution to the nonlinear system (39) and (40) in all numerical examples, although I cannot analytically

prove its existence and uniqueness.

22

as follows. For x < xH , I have VH (x) > aH π(x) − I, and hence(aH π(x) − I)λH /λ̄ + VH (x)λL /λ̄ > aH π(x) − I≥ aL π(x) − I.

(42)

By A2,H > 0 in (8), I have, for x ≥ xH ,(aH π(x) − I)λH /λ̄ + VH (x)λL /λ̄()λH I
λH (aH − 1)π(x)−λL /λ̄
>(aH π(x) − I)λH /λ̄ + π(x) +r + λH − µr + λH)(λH (r + λ̄)I∆aλH
λL /λ̄ −≥aH π(x)λH /λ̄ + aL −λLλ̄(r + λH )λH (r + λ̄)I=aL π(x) −λ̄(r + λH )>aL π(x) − I,

(43)

(44)

where (43) follows from (31). By (44) and (42), I have (41). I also have (aH π(x) − I)λH /λ̄ +U (x)λL /λ̄ ≥ aL π(x) − I for all x ≥ 0 because U (x) ≥ VH (x). Then, the firm value under thehigh-price strategy dominates the payoﬀ by low-price sellout, and U (x) = VH (x) holds.
Next, assume that (31) does not hold. From (8), (aH π(x) − I)λH /λ̄ + VH (x)λL /λ̄ < aL π(x) − I
when x is suﬃciently large. Hence, I have U (x) ̸= VH (x), meaning that the startup posts a low
price. Assume that value function U (x) exists. As in Appendix B, I can derive the value function
U (x) and thresholds x∗∗i (i = H, L), and hence omit the derivation details. By solving ODEs (22)–(24) with boundary conditions (25)–(28), I have Proposition 3, where coeﬃcients B̃i (i = 1, . . . , 7)
are defined by−βB̃1 = (x∗∗((aH − 1)π(x∗∗H)
H ) − I),

B̃2 =

γλ H
γλ H
(x∗∗
B̃6 − (x∗∗
B̃5
H)
L )
,
βλH
∗∗ )γλH − (x∗∗ )βλH (x∗∗ )γλH
(x∗∗
)
(x
L
H
H
L

B̃3 =

βλH
βλH
B̃6
(x∗∗
B̃5 − (x∗∗
H)
L )
,
βλH
γλH
γλ H
∗∗
∗∗
∗∗
(xL )
(xH )
− (xH )βλH (x∗∗
L )

−γλ̄
B̃7 ,
B̃4 = (x∗∗
L )

B̃5 =

(aH − 1)x∗∗rIH−,r + λH − µr + λHλH (aH − 1)π(x∗∗
rI
L )
−
,
r + λH − µr + λHλ̄(aL − 1)π(x∗∗
rI
L )
−
,
B̃7 = (aL − 1 − ∆aλH/λL )π(x∗∗L )−r + λ̄ − µr + λ̄

B̃6 = (aL − 1 − ∆aλH /λL )π(x∗∗
L )−

∗∗
14
and (x∗∗toH , xL ) becomes a solution

λH (aH − 1)π(x∗∗
βλH
γλ H
H)
+ βλH B̃2 (x∗∗+ γλH B̃3 (x∗∗= β((aH − 1)π(x∗∗H)H)
H ) − I),r + λH − µλ̄(aL − 1)π(x∗∗λH (aH − 1)π(x∗∗
βλH
γλ H
L )
L )
+ βλH B̃2 (x∗∗+ γλH B̃3 (x∗∗=+ γλ̄ B̃7 .
L )L )r + λH − µ
r+λ−µ
14

(45)
(46)

I found a unique solution to the nonlinear system (45) and (46) in all numerical examples, although I cannot

analytically prove its existence and uniqueness.

23

D

Proof of Corollary 1

If condition (31) is satisfied, I readily have W (x) = 0. Assume that (31) does not hold. Now, W (x)satisfies ODEs
µxW ′ (x) + 0.5σ 2 x2 W ′′ (x) = rW (x)(0 < x < x∗∗
H ),

(47)

∗∗
µxW ′ (x) + 0.5σ 2 x2 W ′′ (x) + λH (−W (x)) = rW (x) (x∗∗H < x < xL ),(48)µxW ′ (x) + 0.5σ 2 x2 W ′′ (x) + λ̄(∆aπ(x)λH /λ̄ − W (x)) = rW (x)(49)(x > x∗∗L ),with boundary conditions
lim W (x) = 0,(50)

lim W (x)/x < ∞,

(51)

x→0

x→∞

and W (x) continuously diﬀerentiable at thresholds x∗∗i (i = H, L). In ODE (48), λH (−W (x))
corresponds to W (x) becoming 0 with probability λH dt in infinitesimal time interval dt in region∗∗x∗∗
H < x < xL , whereas λ̄(∆aπ(x)λH /λ̄ − W (x)) in ODE (49) corresponds to W (x) becoming∆aπ(x)λH /λ̄ (information rent) with probability λ̄dt in infinitesimal time interval dt in regionx > x∗∗
L . The general solutions to ODE (47) with boundary condition (50), and ODEs (48) and (49)∗∗∗∗∗∗with boundary condition (51) become (34) for x < x∗∗
H , x ∈ (xH , xL ), and x > xL , respectively.

By the continuous diﬀerentiability of W (x) at thresholds x∗∗i (i = H, L), I haveβ∗∗ βλHγλ H
D1 (x∗∗+ D3 (x∗∗,H ) = D2 (xH )
H)

(52)

β
∗∗ βλHγλH
D1 β(x∗∗+ D3 γλH (x∗∗,H ) = D2 βλH (xH )
H)

(53)

βλH
γλH
D2 (x∗∗+ D3 (x∗∗
=
L )
L )

λH ∆aπ(x∗∗L )γλ̄+ D4 (x∗∗L ) ,r + λ̄ − µλH ∆aπ(x∗∗
γλ̄
βλH
γλ H
L )
+ D4 γλ̄ (x∗∗D2 βλH (x∗∗+ D3 γλH (x∗∗
=
L ) .
L )
L )
r + λ̄ − µ

(54)
(55)

By (52) and (53), I haveβλH
γλ H
D2 (βλH − β)(x∗∗+ D3 (γλH − β)(x∗∗
= 0.
H)
H)

(56)

By (54) and (55), I haveβλH
γλ H
D2 (βλH − γλ̄ )(x∗∗+ D3 (γλH − γλ̄ )(x∗∗
=
L )
L )

(1 − γλ̄ )λH ∆aπ(x∗∗L ).
r + λ̄ − µ(57)

By (56) and (57), I have
D2 =

γλH
(γλH − β)(x∗∗(1 − γλ̄ )λH ∆aπ(x∗∗H)
L )D5 (r + λ̄ − µ)D3 = −βλH(βλH − β)(x∗∗(1 − γλ̄ )λH ∆aπ(x∗∗H)
L )D5 (r + λ̄ − µ)

β λH
γλ H
βλH
γλH
D5 = (βλH − γλ̄ )(γλH − β)(x∗∗(x∗∗− (βλH − β)(γλH − γλ̄ )(x∗∗
(x∗∗
.
L )
H)
H)
L )

Coeﬃcients D1 and D4 immediately follows from (52) and (54).

24

(58)
(59)
(60)

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26

Table 1: Baseline parameter values
r

µ

σ

λH

λL

aH

aL

I

x

0.08

0.03

0.18

0.125

0.375

1.5

1.4

10

1

20
18
16
Sym: high price
Asym: high price14

Sym: flexible
Asym: high price

1/ L

12
10
8

Sym: flexible
Asym: flexible

6
4
Baseline

2
0
0

5

10

15

20

1/ H

Figure 1: Strategy choice with respect to expected waiting time 1/λi (i = H, L). The other parametervalues are set as in Table 1. With symmetric information, the firm takes the high-price strategy for
1/λH ≤ 5, whereas with asymmetric information, the firm takes the high-price strategy in the areaon the left-hand side of the blue dashed line. In the region between the black and blue dashed lines,
asymmetric information changes the sellout strategy.

27

40

2

xH

xL

30

xH

1.8

xL

20
1.6
10
1.4

0
5

10

15

20

5

10

1/ H

15

20

15

20

1/ H
1

23
V(x)
U(x)

22.8

0.9

22.6
0.8
22.4

PH (x)

0.7

22.2

QH (x)

0.6

22
5

10

15

20

5

1/ H

10

1/ H

0.24

0.01
S(x)
T(x)

0.22

L(x)

0.008
0.006

0.2
0.004
0.18

0.002

0.16

0
5

10

15

20

5

1/ H

10

15

20

1/ H

Figure 2: Comparative statics with respect to expected waiting time for high type, 1/λH . The
other parameter values are set as in Table 1. The figure plots the sellout thresholds x∗i , x∗∗i , firm
values V (x), U (x), high-price sellout probabilities PH (x), QH (x), state prices S(x), T (x), and social
loss L(x).

281.506

40
xH

xL

30

xH

1.504

xL

20

1.502

10
1.5
0
5

10

15

20

5

1/ L

10

15

20

1/ L
1

22.384V(x)U(x)

22.3820.98
22.38
22.3780.96

PH (x)

22.376

QH (x)

22.374
0.94
5

10

15

5

20

1/ L

10

15

20

1/ L
8

0.1995
S(x)
T(x)

10 -3
L(x)

6

0.199

4
2

0.1985

0
5

10

15

20

5

1/ L

10

15

20

1/ L

Figure 3: Comparative statics with respect to expected waiting time for low type, 1/λL . The otherparameter values are set as in Table 1. The figure plots the sellout thresholds x∗i , x∗∗i , firm values
V (x), U (x), high-price sellout probabilities PH (x), QH (x), state prices S(x), T (x), and social loss L(x).

29

40
1.6

xL

xH
xH

1.5

xL

30
20

1.4

10

1.3
1.2

0
0.02

0.03

0.04

0.05

0.02

0.03

0.04

0.05

1

40

V(x)
U(x)

35

0.98

30

0.96

25

PH (x)

0.94

QH (x)

20
0.92
0.02

0.03

0.04

0.05

0.02

8
S(x)
T(x)

0.3

0.03

0.04

0.05

10 -3
L(x)

6

0.25
4
0.2
2
0.15
0
0.02

0.03

0.04

0.05

0.02

0.03

0.04

0.05

Figure 4: Comparative statics with respect to growth rate µ. The other parameter values are set asin Table 1. The figure plots the sellout thresholds x∗i , x∗∗i , firm values V (x), U (x), high-price sellout
probabilities PH (x), QH (x), state prices S(x), T (x), and social loss L(x).

30

1.6

25

xH
xH

1.55
1.5

15

1.45

10

1.4
0.12

xL

5
0.14

0.16

0.18

0.2

0.22

0.12

22.8

0.14

0.16

0.18

0.2

0.22

1
V(x)
U(x)

22.6

0.96

22.2

0.94

0.14

0.16

PH (x)

0.98

22.4

22
0.12

xL

20

0.18

0.2

0.92
0.12

0.22

QH (x)

0.14

0.16

0.18

0.2

0.22

0.18

0.2

0.22

0.015

0.22

L(x)

S(x)
T(x)

0.21

0.01

0.2
0.005

0.19
0.18
0.12

0.14

0.16

0.18

0.2

0
0.12

0.22

0.14

0.16

Figure 5: Comparative statics with respect to volatility σ. The other parameter values are set asin Table 1. The figure plots the sellout thresholds x∗i , x∗∗i , firm values V (x), U (x), high-price sellout
probabilities PH (x), QH (x), state prices S(x), T (x), and social loss L(x).

31

0.1
Sym.
Asym.

1/ H =10
0.09
1/ H =80.08

1/ H =6

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0
0

5

10

15

20

25

30

35

40

X(t)

Figure 6:

Jump sizes at the sellout time for 1/λH = 6, 8, and 10.

ter values are set as in Table 1.

The other parame-

The black lines plot the symmetric information cases, i.e.,(aH π(X(t)) − I − V (X(t)))/V (X(t)) for X(t) ∈ [x∗H , x∗L ) and (āπ(X(t)) − I − V (X(t)))/V (X(t))
for X(t) ∈ [x∗H , x∗L ), while the blue dotted line plots the asymmetric information cases, i.e.,∗∗(aH π(X(t)) − I − U (X(t)))/U (X(t)) for X(t) ∈ [x∗∗H , xL ) and (aL π(X(t)) − I − U (X(t)))/U (X(t))

for X(t) ∈ [x∗H , x∗L ).

32