Using a Soft Deadline to Counter Monopoly *Masahiro Yoshida†
August 28, 2023

Abstract
A monopolist often exploits a hard deadline to raise their commitment power. I explore
whether a group of buyers can employ a soft deadline to counter the monopoly. Using asimple durable goods monopolist model under a deadline, I show that the buyers’ imperfectcommitment to an earlier exit may elicit a compromise from the monopolist and generate
the buyers’ premium. The soft deadline partially restores the self-competition dynamics
of Coase conjecture, which is previously constrained by the hard deadline. Only one soft
deadline breaks the conventional link between the time horizon (or durability of goods)
and monopoly power.
JEL Classification: C78, C91
Keywords: bargaining, durable goods monopoly, commitment, Coase conjecture, deadline effect* I thank my advisor, Michihiro Kandori, for initiating this project. I have benefited from discussions with
Mariagiovanna Baccara, Marcus Berliant, Junnan He, Hitoshi Matsushima, John Nachbar, Daisuke Oyama,
Jonathan Weinstein and participants at the Midwest Theory Conference at Ohio State, Applied Microeconomic
Theory Workshop at Osaka, and Micro Workshop at Waseda. I thank Fangyuan Yi for her excellent research
assistance. I gratefully acknowledge the scholarship provided by the Nakajima Foundation and McDonell International Scholar’s Academy. The author declares that he has no relevant financial interests that relate to the research
described in this paper.
† Department of Political Science and Economics, Waseda University, Nishi-waseda 1-6-1, Shinjyu-ku, Tokyo,169-8050, JAPAN (e-mail: m.yoshida@waseda.jp)11

Introduction

Across the developed economies, the market power of leading firms is rising. (See e.g. De Loecker
and Eeckhout (2018) for the global evolution of market power and Autor et al. (2020) for therise of superstar firms in the U.S.) Bilateral bargaining typically forms their mode of trade fromprocurement contracts, business partnerships, corporate acquisitions and labor group disputes.
For negotiators in the industrial economy, reinforcing their bargaining power against monopolists is of central importance. Observing that most influential modern monopolists sell durable
goods (e.g. software, intellectual property, semiconductors, natural resources), a workhorse
framework well-suited for the world is a durable goods monopolist model (Stokey (1981); Bulow (1982)). In this scenario, the monopolist sells his goods to a demand pool of buyers with
their private value. The model is widely applied to outside the seller-buyer trades; labor groupdisputes (Hart (1989)), medical malpractice disputes (Sieg (2000)), sovereign debt renegotiation (Bai and Zhang (2012)), and hostage-taking negotiation with pirates (Ambrus, Chaney
and Salitskiy (2018)). Behind the framework, the long-known Coase conjecture (proposedby Coase (1972), later formalized by Gul, Sonnenschein and Wilson (1986)) lies as the theoretical cornerstone. The theory essentially states that the monopolist loses bargaining power
via self-competition when the buyer rationally expects his future concession from the irresistible temptation for price discrimination. Substantial theoretical attempts have been madeto revive the monopolist’s commitment, by depreciating goods (Bond and Samuelson (1984)),
discrete demands (Bagnoli, Salant and Swierzbinski (1989)), the arrival of new buyers (Fuchs
and Skrzypacz (2010)), and the buyer’s outside options (Board and Pycia (2014)).
One natural tactic to circumvent the conjecture is commitment to a deadline (Sobel and
Takahashi (1983); Fudenberg, Levine and Tirole (1985)). Casual observation suggests thatmonopolists maintain a stronger reputational concern by imposing a perfectly committed harddeadline. Armed with the deadline, even as the proposer’s commitment to the offer disappears(i.e., a length of each bargaining round), the proposer regains his bargaining power by framing
the offer as an “ultimatum”. (See, e.g., Fershtman and Seidmann (1993); Güth and Ritzberger(1998))1 In this paper, I question whether a deadline is an exclusive commitment device forthe monopolist. Each buyer’s reputational concern is plausibly weaker than the monopolist.
1 Gneezy, Haruvy and Roth (2003) consider an infinite-horizon version of ultimatum game where the proposercompletely loses the bargaining power, and call it a “reverse” ultimatum game. However, the simple impositionof the deadline completely recovers the power, making the game resemble a canonical ultimatum game.

2

I therefore explore whether a group of buyers could partially commit to an intermediate soft
deadline to counter the monopoly power recovered by the hard deadline under the classicalcontext of a durable goods monopoly.
Soft deadlines have been implemented or may well be implemented among a variety of collective bargaining disputes in the real world. For instance, in a procurement for scarce resources(e.g. gasoline, electricity, natural resources, semiconductors), a group of small-sized firms
or municipalities very often form a purchase consortium to negotiate with a monopolistic—occasionally foreign—supplier.2 (Regarding purchasing consortia, see, e.g. Glock and Hochrein(2011); Tella and Virolainen (2005). For collaborative procurement in the public sector, see
Walker et al. (2013).) This consortium of small firms might reinforce their bargaining powerby partially committing to earlier switching to another supplier. In a labor union negotiation, aunion bargains over a wage under a publicly announced deadline regarding when a strike starts(see, e.g. Card (1990); Cramton and Tracy (1992)). The union may dangle the possibility ofan earlier strike, conditional on a sluggish negotiation progress.3 In another example, in negotiations to release the captured hostages, terror groups or pirates often demand the paymentof ransom within a deadline.4 The government or the police serving in a diplomatic role often
issue a warning of earlier suppression by force before the deadline. I formally illustrate thesescenarios by naturally extending a simple Coasian model under a deadline and show that thebuyers’ imperfect commitment to an earlier exit would create a novel motive for price discrimination, and thus eliciting a concession from the monopolist.
I start with the canonical bargaining model of a durable goods monopolist. Consider thata monopolist (he) sells his goods of zero marginal costs to a continuum of buyers (she) and
imposes a perfectly committed hard deadline at time T . A single buyer’s valuation is private
information, but the value distribution is shared as common knowledge. Each buyer rejects
2 A large conglomerate firm sometimes runs a collective procurement instead of an independent purchase by

its subsidiary. Under the recent global shortage of semiconductors, for example, Toyota (a large auto company)
launched a group purchase (Daily Industrial Newspaper, April 21, 2023).
3 Aside from the industrial world, a soft deadline commitment has been applied in the international sovereign
debt negotiations. Greece held a national referendum before the default deadline to accept the bailout proposals by
the EC, ECB and IMF in 2015. Although it entailed no legal binding power, the analysts predicted that Greece’s
rejection might trigger a chain reaction of financial terror. (The Teregraph, July 6, 2015)
4 Ransom remains to be a vital source of funding for terrorist groups and pirates. For example, the terror group
al-Qaeda obtained 125 million dollars in ransom from kidnapping during 2008–2014 (New York Times, July 30,2014). In addition, pirates in Somalia received 360 million dollars during 2005–2012 (Yikona (2013)). A dataset
by Mickolus et al. (1976) records the requested ransom and imposed deadlines in the history of terrorist incidents.

3every offer until the offer is accepted. When the deadline passes, both get 0. In the uniqueequilibrium, the asymmetric information generates a delay until agreement to screen the buyer;
the price schedule is declining overtime, and a lower-type buyer wait longer for a discounted
price.
In this environment, monopolistic power is hindered by two forces. First, the monopolist
performs an intertemporal price discrimination endowed with multiple rounds of price offers.
Second, the monopolist expects the buyer’s incentive to wait and discounts the price under the
strategic interaction. Consequently without a deadline, self-competition occurs, as conjectured
by Coase (1972).5 A hard deadline constraints the both forces: When the deadline approaches,a buyer is less likely to wait. Thus, the monopolist’s offer resembles an ultimatum.
Suppose that before the bargaining starts, a group of buyers announces a soft deadline ata specific earlier time before the hard deadline, which I call a threat point just after the timet ∗ (see Figure 1). To maximize the expected surplus aggregated from the demand pool, theFigure 1: soft vs. hard deadline
Note: A soft deadline with a conditional probability of breakdown α ∈ (0, 1) is imposed at time t ∗ in contrast to
the hard deadline at time T . The buyer’s group commits that the bargaining might end at t ∗ with probability α but
continue with probability 1 − α. In the main text below, I allow for multiple soft deadlines for generality.

group publicly announces its level of commitment to the deadline, characterized by a conditional probability of exit α, if the bargaining exceeds that point. My soft deadline frameworknests a hard deadline setting as α = 0 or 1. Intuitively, this soft deadline serves as an uncertain“time bomb” where the lower-type buyers who have not yet agreed to an offer would stochastically leave the bargaining table. (i.e., fall back to their outside options 0). One may view
this commitment technology as an application of uncertain commitment (initially proposed by
5 Güth (1994) called the dynamics as the intrapersonal price competition of the monopolist. In the language ofscreening-type bargaining, if an uninformed proposer has more frequent opportunities to revise the offers, and theinformed responder rationally expects a future concession, the proposer loses the commitment power of offers.

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Crawford (1982), and later, investigated by Ellingsen and Miettinen (2008)) to the deadline.
Consistent with the literature with a hard deadline with deadline effect (as tested in Roth,
Murnighan and Schoumaker (1988) and formalized in Fershtman and Seidmann (1993)), I firstshow that more agreements are likely to occur just before the soft deadline.6 This compromiseon the buyers’ side is framed as a stochastic analog of deadline effect. Within the strategicinteraction, the monopolist exploits the buyers’ compromise in an analogous way as an ultimatum game. More intriguingly, however, I show that the surprisingly subtle imposition of
the soft deadline elicits the monopolist’s price discrimination in a non-obvious way, but withintuitive appeal.
When the soft deadline safely passes (t > t ∗ ), I show that the monopolist sharply performs
a compromise, characterized as an atom of price cut. (See Figure 2.) This major sale is a new
result of my model; occurring because if the buyer keeps rejecting at the threat point, she credibly signals that her value is substantially low. This is a direct consequence from the buyers’
compromise at the threat point. Rejection at the soft deadline signals that agreement before
the “time bomb” is still not profitable enough for the remaining buyer, and the monopolist is
tempted to price discriminate. After the big sale, a remaining buyer benefits from the monopolist’s discount. Expecting the less lucrative market after the soft deadline, the forward-looking
monopolist may respond by driving down the price schedule (t ≤ t ∗ ) with a cheaper opening
price at t = 0 to raise the probability of agreement before the soft deadline. (See Figure 2.)
One may view a soft deadline as partially restoring the self-competition dynamics of Coase
conjecture previously constrained by the hard deadline.7
At the soft deadline, both the compromise from price discrimination and exploitation fromstrategic interaction are at work. As both effects are counteracted, I formally demonstrate thatthe buyers’ reasonably imperfect commitment potentially yields an aggregate premium. If thecommitment is too hard, however, the soft deadline resembles a hard deadline, and the normal
strategic interaction dominates and just backfires to augment the monopolist’s power. I showb ∈ (0, 1) to the deadline, maximizing thethat there exists an optimal interior commitment α
6 In a market with a continuum value of buyers, this phenomenon is typically interpreted as a disproportionate

agglomeration of purchases.
7 In contrast to the Coase conjecture, the Packman conjecture (see, e.g. Bagnoli, Salant and Swierzbinski(1989)) claims that price discrimination is a source of monopoly power. Although this theory holds true for otherbargaining protocols (e.g. complete information. See e.g. Von der Fehr and Kühn (1995)), this paper adopts
Coase’s view that myopic price discrimination undermines ex-ante monopoly power by hurting the commitment
to pricing.

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expected surplus of the buyers. Interestingly, under some parameter values, buyers of all types
are shown to be better-off because of, or at least indifferent to, the imperfect commitment,
assuring a participation constraint for a group to impose this commitment. (See Figure 4)
The finding casts a new light on the conventional wisdom regarding the bargaining horizon(or the typical interpretation as the durability of goods) and monopoly power under the durable
goods monopoly. The Coase conjecture claims that durability reduces monopoly power, indicating that as the bargaining horizon expands, the surplus division is in favor of the buyers andthe market becomes more efficient.8 To see this claim intuitively, consider two polar cases.
Under a one-shot time, the game is reduced to an ultimatum game with one-sided incomplete
information.9 The buyers suffer most under the strongest monopolist’s commitment. In contrast, in an infinite horizon scenario, the monopolist is endowed with limitless opportunities
of offers. Buyers enjoy competitive pricing and the bargaining is Pareto-efficient aligned with
the Coase conjecture. In a more generic time structure allowing for soft deadlines, the finding
suggests that bargaining appears shorter, but the buyers are better off in expectation.
Literature review A huge body of the theoretical bargaining literature has proposed the tactics to augment bargaining power, such as securing outside options (Compte and Jehiel (2002);
Fuchs and Skrzypacz (2013); Hwang and Li (2017)), cheap talk (Crawford and Sobel (1982).
See Farrell and Rabin (1996) for a survey.), and reputation from strategic irrationality (Abreu
and Gul (2000); Kambe (1999)). Most relevant to my study is the literature on commitment
initially ideated by Schelling (1960), and later formalized by Crawford (1982), Muthoo (1996),
and Ellingsen and Miettinen (2008). Much of the literature stochastically frames credibilityof commitment on the proposer’s offer with some probability (α in my model), and very few
examines the use of a deadline as a commitment technology.
The paper is also related to bargaining models with a strategic use of a given deadline.
(Ma and Manove (1993); Fershtman and Seidmann (1993)). These studies chiefly explore theproposer’s manipulation of delaying an offer with an intention to frame it as an ultimatum.10
8 See, e.g., Güth and Ritzberger (1998).
9 The ultimatum game is a two-stage game where a proposer and a responder bargain over a fixed money. In

the first stage, the proposer offers his share, and in the second stage, the responder accepts or rejects it. If it isaccepted, each receives money based on the offer; if rejected, each gets 0. A key difference from my model is thatthe game has complete information.
10 The intuition echoes the revenue management literature (e.g., Horner and Samuelson (2011)), where thedurable goods are perishable at the deadline. The key difference is that they limit the stocks of the goods, whereasa durable goods monopolist model is not subject to such limits.

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Instead of strategic timing of the offer, my model adds the possibility of an exit at a specific
event before the deadline. Using the reputation approach with the war of attrition protocol ofKambe (1999), Özyurt (2023) jointly analyzes the commitment to the offer of the proposer andthe endogenous timing of the deadline, showing that the deadline setter is better off under theefficient unique equilibrium. This paper features the credibility of the deadline instead of its
timing.
This study also builds on theoretical bargaining works on the role of hard deadlines on
last-minute agreements (or the “deadline effect”). In the context of pre-trial negotiations, Spier(1992) uses a one-sided incomplete-information model similar to mine, deriving an agglomeration of trade at the deadline of a trial date. Ponsati (1995) and Damiano, Li and Suen (2012)derived a similar atom of trade at the hard deadline in concession games with two-sided incomplete information. My paper demonstrated that deadline effect occurs even if the deadline issoft. Fanning (2016) uses the reputation model of Abreu and Gul (2000) to provide a foundationof deadline effects from reputation across a wide range of protocols. In his model, however,
one-sided incomplete information in the durable goods monopoly generates no delay.11
The main result of my paper (Proposition 4) contributes to the classic literature on the
durable goods monopolist model (Stokey (1981), Bulow (1982), Gul, Sonnenschein and Wilson (1986)). Most of the literature adopts an infinite horizon framework. The avoidance ofthe zero-profit trap in the finite horizon framework is contained by Stokey (1979), Sobel and
Takahashi (1983), and Fudenberg and Tirole (1983), although the role of a deadline is not explicitly mentioned. 12 Real-world bargainings often entails deadlines, and a series of applied
works adopt a finite-horizon framework (e.g. labor union disputes for Tracy (1987), medical
malpractice disputes for Sieg (2000), sovereign debt renegotiation for Bai and Zhang (2012)).
All the durable goods monopoly literature in theoretical or applied works inherits the accepted
wisdom launched in a seminar paper Coase (1972) titled “durability and monopoly”, indicatingthat a shorter horizon (or less durability of goods) augments the monopoly power.13 The key
novelty is that adding only one soft deadline would break the conventional relationship between
11 See the “Related Literature” section in Fanning (2016) for a comparison with one-sided, incomplete informa-tion models (e.g. Spier (1992); Fuchs and Skrzypacz (2013)).
12 Ostatnickỳ (2004) emphasizes that a hard deadline fails the Coase conjecture, exploring the effect of discounting and decreasing private value to partially recover the Coase conjecture.
13 Later studies formalize the relationship. (e.g.; Sobel and Takahashi (1983), Theorem 6, or Güth and Ritzberger(1998)). In the infinite horizon model, Stokey (1981) delivers a similar intuition; a longer time between offers (orless frequent offer revision) leads to a monopolist’s gain.

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the time horizon and monopoly power.
Though outside the durable goods monopoly context, the closest precedent to using a deadline in the continuous time is Fuchs and Skrzypacz (2013). In general, solving finite horizon
models in the continuous limit is technically challenging, because the agent faces different time
horizons at each decision under a non-stationary time structure. For the sake of tractability, I
borrow their proof technique, rooted in Ausubel and Deneckere (1992). Conceptually, my paper diverges from Fuchs and Skrzypacz (2013) in three crucial dimensions. First, Fuchs andSkrzypacz explore the effect of relative outside options after a hard deadline on deadline effects, while my model highlights the role of a soft deadline on monopoly power, standardizing
the outside options. Second, I show that a deadline effect also exists in a more general deadline
structure. Third, my model delivers the new prediction of an atom price drop as a result of
emerging price discrimination.
The characterization of soft deadlines is isomorphic to the idea of a random breakdown.
Binmore, Rubinstein and Wolinsky (1986) introduced the risk of breakdown in the alternatingoffers model a la Rubinstein (1982). Rubinstein and Wolinsky (1985) embed the risk in themodel of decentralized market. The early models include no asymmetric information, andthus, generating no costly delays in equilibrium. However, later studies build models withincomplete information and random breakdown, where the timing of breakdown is uncertain
under a continuous deadline distribution (Fuchs and Skrzypacz (2010); Fanning (2016); Simsek and Yildiz (2016)). By contrast, my model presumes that an occurrence of breakdown is
uncertain for a specific event, contrived to capture a realistic feature of bargaining institutions.
If breakdown comes with a continuous arrival rate, the magnitude of agglomeration of trades(deadline effects) or the price drop cannot be measured by atoms as in my paper (See Lemma
1 and Proposition 2).
On the technical front, a random breakdown is typically implemented as changing discounting factors. This formulation is also adopted by models with a stochastic arrival of deadlines
in the continuous time, where a discount rate is adjusted by an arrival rate of a deadline. Mathematically, a discount factor is adjusted in my discrete period model to capture the credibilityof the soft deadline. Meanwhile, in the limit case, a discount rate is not adjusted. Most of thebargaining literature deviating from the solo discount factors explores how the relative patienceof both parties alters the surplus division (see, e.g., Binmore, Rubinstein and Wolinsky (1986);

8

Chapter 4.4 in Muthoo (1999)).14 One could also view that my soft-deadline framework presumes a common patience for both parties, while allowing for more dynamically-varying patience (e.g. hyperbolic discounting).
Outline The paper is organized as follows: Section 2 presents a bargaining framework with
an imperfectly committed deadline and characterizes the unique equilibrium. Section 3 showsthat the monopolist performs a compromise before and after the threat point. Next, Section 4
demonstrates that the buyer’s side enjoys a premium. The sensitivity of the overall efficiency
is explored regarding the commitment intensity of the buyer’s side. Section 5 concludes thepaper. Most of the proofs of the results are provided in the Appendix.

2

Model

I start with a durable goods monopolist model, where an uninformed monopolist screens a poolof buyers under a hard deadline. I state the results in continuous time for simplicity althoughmost of the results in the discrete period version remain unchanged.

2.1

Setup

A monopolist (he) negotiates with a buyer (she) to sell an indivisible good. Both are risk-neutral
and forward-looking expected utility maximizers. The buyer has her private value v ∈ [0, 1] forthe good and I assume that v is distributed according to the shared cumulative distributionfunction F(v) = vσ (σ > 0).15
The monopolist’s marginal cost is normalized to 0, and it is common knowledge.16 Time iscontinuous with t ∈ [0, T ] and broken into a length ∆ > 0 for each bargaining round. Supposethat the monopolist credibly imposes a hard deadline at the time T . At the beginning of thetime t, the monopolist proposes an offer pt .17 Then, the buyer immediately either accepts or
14 For a durable goods context, Guth and Ritzberger (1992) explore that the relative patience of monopolists vs.

consumers shapes the surplus division.
15 This distributional assumption yields two benefits. First, the distribution shape is unchanged with Bayesian
updating, which offers closed-form solutions by solving backwards. (Ausubel and Deneckere (1992), Fuchs
and Skrzypacz (2013)) Second, it captures the first-order stochastic dominance. I conjecture that the qualitative
predictions of the model are insensitive for any atomless and full-support distributions.
16 This corresponds to the “no-gap” case, where a marginal cost is no lower than the lower bound of the buyer’s
private value.
17 Ausubel and Deneckere (1992) provides a justification for the rule in which only the uninformed party is per-9

rejects.18 If she accepts the price at the time t, the game ends with this outcome: the monopolist
gets e−γt pt , and the buyer gets e−γt (v − pt ), where γ is a common instantaneous discount rate.
If she keep rejecting the price until t = T , both get 0 as an outside option.
Suppose that the buyers form a group to maximize their expected payoff, and the group
imposes a series of soft deadlines at an earlier time td ∈ (0, T ) (d ∈ {1, · · · , N}) with someimperfect commitment, captured by a conditional breakdown risk αd ∈ (0, 1) at the end of thetime td .19 When N = 1, I omit d for simplicity and write it as t ∗ . (e.g. Figure 1) This implies
that if the proposal is rejected at time td , the bargaining ends with probability αd and both get
0, but it proceeds to time td + ∆ with probability 1 − αd .20 Observe that if αd = 1 for some d,the soft deadline is reduced to a hard deadline.

2.2

Equilibrium

What follows is a straightforward application of the canonical bargaining with incomplete information. (See Fudenberg, Levine and Tirole (1985) for the theoretical foundation.) Let [0, kt )be a posterior valuation at time t, and both players share {kt } as a belief system. Then, the buyer
adopts this specific form of cutoff strategy: given kt and pt , she calculates the cutoff value ct ,satisfying the necessary condition for optimality:21
ct − pt| {z }

payoff of agreement today=

ηt e−γ∆ (ct − pt+∆ ) .
|{z}payoff of agreement tomorrow

Intuitively, the marginal buyer with a value v = ct is indifferent between buying today or
tomorrow. Then, at time t, the buyer accepts pt if v ≥ ct , and rejects if v < ct . Immediately fromthis cutoff strategy, the belief system for both players is characterized by the cutoff schedule

ct = kt+∆ (∀t ∈ [0, T − ∆]),

(1)

mitted to make offers. They showed that under alternating-offer games with one-sided incomplete information, theinformed party endogenously never makes any serious offers when ∆ is sufficiently short (the Silence Theorem).
18 They are allowed to use mixed strategies, but this does not change the argument because the buyer’s mixed
strategy is rationalizable only when the private value is equal to the cutoff.
19 Their participation constraint for the group is discussed below.
20 This is different from cheap talk (see Farrell and Rabin (1996) for a survey) in that the group commits to therealization of breakdown. The setting is in line with bargaining with breakdown (e.g., Rubinstein and Wolinsky(1985); Binmore, Rubinstein and Wolinsky (1986)) as discussed in the literature review.
21 Note that c is a function of p , not of v. Also, this necessary condition turns out to be sufficient due to thettskimming property (i.e., the higher types agree earlier than the lower types). One can check that the difference
between both hands is strictly increasing in ct .

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suggesting that the cutoff today is a supremum value of the buyer tomorrow. Then, theproblem of the monopolist and the buyers’ group is recursively defined as follows:
Vt = max (pt|

F(kt ) − F(ct )) pt +
F(kt ){z}probability of agreement

Wt = max (ct|

F(ct )F(k )| {zt }

ηt e−γ∆Vt+∆(2)

probability of rejection

F(kt ) − F(ct )) {E(v | ct ≤ v < kt ) − pt } +F(kt ){z}probability of agreement

F(ct )F(k )| {zt }

ηt e−γ∆Wt+∆(3)

probability of rejection

where Vt and Wt are value functions of each party, 22 and ηt is a risk adjustment factor
attached to a discount factor e−γ∆ such that

ηt =



1 − αd(t = td )1(t 6= td ).

(4)

Mathematically, this is nothing but adjusting a discount factor in the particular time points
td . I show below that this surprisingly simple formulation of commitment intensity regarding
the deadline generates a perhaps unintended consequence for surplus division.23 Lastly, anequilibrium is defined as a standard perfect Bayesian equilibrium.
Definition 1. A pair of strategies {(pt , ct )} and a belief system {kt } constitutes a perfect
Bayesian equilibrium of the game if their actions maximize their expected payoffs at all information sets and a belief system is consistent with Bayes rule whenever possible.

2.3

Dynamic Schedules

I solve the model backward. (See Sobel and Takahashi (1983) and the appendix in Fuchs andSkrzypacz (2013)) Given the state variable {kt } at time t, the equilibrium path of {(pt , ct )}(t ∈ [0, T ]) is sequentially characterized by {at } and {bt } as follows:

pt = at kt and ct = bt pt(5)

22W is interpreted as the collective buyer’s continuation value at the time t.
t
23 One can see this protocol adopts a general time discounting, including hyperbolic discounting and non-geometric discounting.

11where {at } and {bt } are recursively characterized by the following difference equations:

−1



at = {(σ + 1) − σ ηt δ at+∆ bt } σ /bt , (t < T )bt = {1 − ηt δ (1 − at+∆ )}−1(t < T ) ,





aT = (1 + σ ) −1
σ ; bT = 1.

(6)

The distribution of F(v) = vσ yields that both pt and ct are linear in kt . Intuitively, at and bt
capture the monopolist’s and the buyer’s bargaining power, respectively. Especially, a0 captures
the ex-ante monopoly power. In equilibrium, regardless of the history of prices, i.e., {p0 , · ·· , pt−∆ }, the monopolist turns out to choose pt only based on kt and the buyer chooses ct
depending only on the current price pt .
Given the strategies of both players, we are ready to characterize the equilibrium.
Proposition 1. [Equilibrium] A unique Perfect Bayesian Equilibrium path of {(pt , ct )} ischaracterized by (4), (5), and (6).
By straightforward induction, value functions of both players are pinned down as follows.
Corollary 1. [Monopolist’s bargaining power and value functions]
The monopolist’s and the buyer’s value functions Vt and Wt , respectively, are characterized
by the monopolist’s bargaining power {at }:

Vt =

σ
σ
σ +2
at kt , Wt = {(1 −at )}kt
σ +1
σ +1
σ +1

Analogous to prices and cutoffs, the value functions of both the monopolist and the buyer
are also linear with respect to the state variable kt , due to the convenience of a functional formof F(v) = vσ . The value functions are used to characterize the ex-ante expected surplus ofeach party: when t = 0, V0 and W0 captures an ex-ante monopolist’s and consumer surplus,
respectively, both of which are linear in the ex-ante monopoly power a0 . The formulations ofsurpluses are utilized in Section 4 and the bargaining efficiency in Section 5.

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3

Price Schedules

In this section, I show that the monopolist as well as the buyer make some compromises dueto the soft deadline. To rigorously characterize the sizes of such compromises, I extend theframework in the limit case as ∆ → 0, following the techniques of Ausubel and Deneckere(1992) and Fuchs and Skrzypacz (2013) under boundary conditions of the intermediate andterminal deadlines.24 Note that even when the monopolist’s stubbornness to each price decays(i.e., ∆ → 0), a costly delay remains under the hard deadline, evading the Coase conjecture.
Given that the bargaining has a delay, how does the threat of exit change players’ behaviors
before and after the threat point? I address this question by Lemma 1 and Proposition 2, jointly
visualized in Figure 2. One can see that both the cutoff and price schedules discontinuously

Figure 2: Bargaining dynamics in equilibrium (baseline vs. optimal commitment)
Note: The schedules of the players are simulated in the continuous-time model, where a soft deadline with α =0.274 is imposed at t ∗ = 3 out of the time horizon T = 6. The bargaining might end at t ∗ = 3 with probability α,
but it might continue with probability 1 − α.

drop around the threat point. As I set ∆ → 0, I analytically pin down the atom of each dropwith bargaining primitives γ, σ , α, T and t ∗ . Before discussing the monopolist’s compromise,
I start by showing the buyer’s compromise at Lemma 1.
Lemma 1. [Deadline effect]
24 See Figure 7 in Appendix for the construction of variables in detail.

13

In the limit as ∆ → 0, the buyer’s cutoff schedule ct is continuous at t ∈ [0,t1 ], t ∈ [td ,td+1 ) (d ∈{1, · · · , N − 1}), and t ∈ (tN , T ], but ct discontinuously drops at t = td (∀ d ∈ {1, · · · , N}).
Moreover, the cutoff drop at t = td is strictly increasing in αd . (∀ d ∈ {1, · · · , N})[Sketch of Proof] Let us denote that the size of a discontinuous cutoff drop at t = td asg(αd ). g(αd ) at t = td (d = 1, · · · , N) is formulated byg(αd ) ≡ lim ct − ctd = (1 − atd btd )ktdt↑td

Note g(αd ) is a function of primitives αd , σ , and limat , which is determined by σ , γ, td , andt↓td

T . One can check that g(αd ) ≥ 0 if and only if αd ≥ 0, and g(0) = 0. (See Appendix for an
explicit form of g(αd ) and derivation.) 

Lemma 1 states that the cutoff schedule sharply drops just before the threat point. In the
durable good market, an instantaneous flow of trade occurs at t ∈ [0,t ∗ ) and t ∈ (t ∗ , T ), but
an atom of trade takes place at t = t ∗ . This is very intuitive: when the buyer faces the “timebomb”, she responds by dropping the cutoff sharply. I also show that the commitment (largerα) monotonically expands the drop. The earlier drop can be viewed as a stochastic analog
of the deadline effect (tested by Roth, Murnighan and Schoumaker (1988) and formalized by
Spier (1992) and Fuchs and Skrzypacz (2013)25 ) Thus, the next result follows directly from
Lemma 1.
Proposition 2. [Sale after the threat point]
In the limit as ∆ → 0, the monopolist’s price schedule pt is continuous at t ∈ [0,t1 ] andt ∈ (td ,td+1 ] for all d ∈ {1, · · · , N − 1}, t ∈ (tN , T ], but pt discontinuously drops at t = td (∀ d ∈{1, · · · , N}). Moreover, the price drop at t = td is strictly increasing in αd (∀ d ∈ {1, · · · , N}).
[Sketch of the Proof] Let us denote that the size of a discontinuous price drop at t = td as
25 Fuchs and Skrzypacz (2013) consider the limit case of finite-horizon bargaining with asymmetric informationand derive an atom trade at the hard deadline.

14h(αd ). h(αd ) is formulated byh(αd ) ≡ ptd − lim pt = atd ktd − limat limkt =
t↓td

t↓td

t↓td

atd ktd(1 − limat ) αd
xt ∗
t↓t
|
{z d }>0

αd + (1 − αd )limat
=(

t↓td

1

) σ (1 − limat )ktd αd .
αd (σ + 1) + limat (1 − αd )
t↓td
|
{z
}
t↓td
independent of αdholds. One can see the price drop h(αd ) ≥ 0 if and only if αd ≥ 0, and h(0) = 0. This observation implies that when αd > 0, the price schedule discontinuously drops at t = td . 

Proposition 2 states that when α ∈ (0, 1), an atom of discount occurs if the bargaining survives
the threat point. (See Figure 2.) Moreover, the magnitude of the price discount expands withthe commitment of the buyer. In fact, this is a direct consequence of the stochastic deadline
effect in Lemma 1. Because the buyer did not accept before the threat point, the buyer’s value isdemonstrably bounded. Put differently, her rejection at the threat point is a costly, but effective
signal of the lowness of her valuation. Given the limited posterior, the monopolist is forced
to sharply discount the price. One may view that the soft deadline serves as a self-screening
device of private information at the cost of expected breakdown. Intriguingly, given his postthreat concession, rational backward induction dictates that the monopolist offers a cheaper
pre-threat price schedule, formally stated as follows.
Proposition 3. [Discount before the threat point]
There exists some α d ∈ (0, 1) such that for αd ∈ (0, α d ], the monopolist offers a uniformly lower
price pt (αd ) < pt (αd = 0) for all t ∈ [0,td ] ∀ d ∈ {1, · · · , N}.

A proof directly comes from Proof of Proposition 4. (See Appendix) A soft deadline (α ∈(0, 1)) makes a monopolist’s cost–benefit accounting qualitatively different from a hard deadline (α = 1); the risk-neutral monopolist weighs securing the current expected payoff by myopically leveraging the buyer’s compromise at t ∈ [0,t ∗ ] against an option value of post-threat
continuation at t ∈ (t ∗ , T ]. At his optimization problem in the face of the threat, observe that a
trade-off emerges between leveraging the buyer’s compromise by raising a price (strategic interaction) vs. securing an agreement by discounting a price (price discrimination). As the threat15

of exit increases, the monopolist projects that the market after the threat point is less lucrative and the post-threat market itself is likely to disappear. Thus, through the self-competition
across before- and after-threat point, he may be tempted to discount an opening price to raise
the probability of agreement before the threat point.

4

Monopoly Power and Consumer Surplus

Buyers’ premium from the soft deadline
Based on the pair of compromises of the monopolist before and after the threat, I present themain result of the study; an optimally designed soft deadline yields a premium to the buyers’
group, achieving their maximum expected surplus.
Proposition 4. [Buyers’ premium from the soft deadline]
cd ∈ (0, 1) that uniquely maximizes W0 s.t.
In the limit as ∆ → 0, there exists α(limat )2
cd =
α

t↓td

(7)

(1 − limat )(1 + σ − limat )
t↓td

t↓td

where limat is recursively characterized by bargaining primitives γ, T,td and αd for d =t↓td

1, 2, · · · , N s.t.
−1aT = (1 + σ ) σ , limat = e−γ(T −td ) atd+1 , and atd+1 = t↓td{αd+1 + (1 − αd+1 ) lim at }1+σ

 σ1

(1 + σ )αd+1 + (1 − αd+1 ) lim att↓td+1t↓td+1

holds.26[Sketch of the Proof] From Corollary 1, W0 =

σ
σ +2
(1 −
a0 ) holds. Differentiating
σ +1
σ +1

W0 with respect to αd , one gets
dW0
σ (σ + 2) da0
σ (σ + 2) da0 datd
=−
=
−
.
dαd
(σ + 1)2 dαd(σ + 1)2 datd dαd

(8)

−1

26 When N = 1, recursive notations are simplified to lima = e−γ(T −t ∗ ) a = e−γ(T −t ∗ ) (1 + σ ) σ .
t
T
t↓t ∗

16

da0
> 0 (shown in Appendix) and analyze the impact of αd on atd .
datd
Differentiating atd with respect to αd , the first-order condition yields
It is sufficient to showdatd (btd , αd )
dαd
|
{z
}

change in monopoly power

datd
dbtd
|{z}

=

dbtd
dαd
|{z}

+

monopolist’s exploitation (< 0) buyer’s compromise (< 0)|strategic interaction=0monopolist’s compromise (< 0)

} |

{z

∂ atd
∂ αd
|{z}
{z

}

price discrimination(9)
Arranging w.r.t. αd , one gets
(limat )2 (αd − 1) + αd {1 + σ − limat (2 + σ )} = 0.
t↓td

t↓td

cd . (The detailed derivation and the second-order condition isSolving for αd , one gets α
shown in Appendix.) This proposition states that a positive, but imperfect commitment to the soft deadline maximizes the consumer surplus. Recall that the ex-ante consumer surplus W0 is decreasing in theex-ante monopoly power a0 (Corollary 1). The ex-ante monopoly power a0 is shaped by themonopoly power at the soft deadline atd (See the dynamics of the monopoly power for an equation (6)). As such, the non-linearity of the consumer surplus stems from two forces working
at the soft deadline: a canonical first-mover advantage of exploitation (strategic interaction;
datd dbtd) and an irresistible compromise to secure the pre-threat agreement (price discriminadbtd dαd∂ atd). It can be shown that as the soft deadline gets harder, the response from strategic
tion;
∂ αd
b , the twointeraction exceeds the one from price discrimination. At the optimal commitment α
responses are balanced.
If the soft deadline becomes hard (α = 1), the optimization is reduced to an ultimatum game∂ atdwithout the factor, and the monopolist exclusively exploits the buyers. Under some mod∂ αderate threat of exit, however, as the buyers rationally foresee that the monopolist is tempted toprice discriminate after the threat point (Proposition 2), he is potentially convinced to concede
at the beginning under the self-competition. One may observe that the motive for price discrimination generates a self-competition dynamics previously restrained under the hard deadline.
As the buyer’s expected surplus is higher than in the non-commitment case (α = 0 or 1),
the buyers’ group enjoys a premium at the cost of expected breakdown. Figure 3 shows the17inverted-U sensitivity of the expected surplus of the demand side.

Figure 3: Commitment to the deadline and the buyer’s expected surplus
Note: A model is simulated with T = 6, σ = 1, γ = − ln(0.98), and a single threat point is set at t ∗ = 3. The
b = 0.274.
vertical line is the optimal commitment α

A simulation under primitives (T = 6, σ = 1, γ = − log(0.98)) finds that the group’s exb = 0.274. With every rise of 10 p.p. of thepected surplus is maximized at an interior risk αb , the expected surplus improves by 6.8 p.p. but otherwise deteriorates byrisk, when α < αb with primitives σ and γ below.)
3.6 p.p. (I provide comparative statics of αWhat does this mean for the durable goods monopolist literature? As is discussed in theliterature review, Coase conjecture negatively relates the durability and monopoly power. The
central logic of the conjecture is self-competition dynamics, where the buyer rationally foresees
the monopolist’s price discrimination and the monopolist loses his commitment on pricing. The
conventional wisdom is that a longer bargaining horizon (or interpretably, less durability) hurt
the monopoly power. In fact, the buyer’s expected surplus under the no-commitment baseline
with a single threat point (N = 1) is characterized by
1+2σ
W0 = E(v) 1 − (2 + σ )(1 + σ )− σ exp(−γT ) ,which is strictly increasing in the horizon T .27 Under the one-shot game (T → 0), the monopolist gains the strongest bargaining power, and the buyer’s surplus is minimized at W0 =1+2σE(v)(1 − (2 + σ )(1 + σ )− σ ). Essentially, the game is reduced to an ultimatum game with
27 See the Appendix for a derivation.

18

asymmetric information. In the infinite horizon (T → ∞), by contrast, W0 increases to E(v),seizing the maximum efficiency as in the Coase conjecture; the marginal-cost trade occurs withno delay. By stochastically mixing a short and long deadline, the model casts new light on the
canonical link between the bargaining horizon and the surplus division; as the commitment toexit increases, the bargaining appears shorter in expectation but is favorable for the buyers.

Cost-benefit analysis across the buyer’s types
Thus far, I have characterized the optimal commitment on the soft deadline strategy, but thecredibility of the soft deadline is a commitment by the buyer’s group. Which type of thebuyer is better off with an optimal commitment? Are there any buyers getting worse off by
the commitment? To see this, I compute the premium across each type of buyer from thecommitment. Figure 4 displays the ex-ante surplus of the each type of buyer (top), and themonopolist’s expected revenue when facing each type of buyer (bottom). Intriguingly with the

Figure 4: Expected surplus across types (optimal commitment (solid line) vs. baseline (dashedline))
Note: A model is simulated with T = 6, σ = 1, γ = − ln(0.98), and a single threat point is set at t ∗ = 3. A shaded
blue or red area is an expected gain from the soft deadline for buyers or the monopolist, respectively.

19

baseline parameter values (T = 6, σ = 1, γ = − ln(0.98)), every type of risk-neutral buyer withv ≥ cT (= 0.26) is strictly better off even after taking the termination cost into consideration.
One can see that the following cost–benefit analysis of the commitment holds for each segmentof buyers. (For notational convenience, denote a hat as the optimal commitment, and 0 as the
no-commitment baseline.28 )• A buyer with v ∈[cct ∗ , 1] is better off with earlier agreement on a cheaper price pt∗ at t ≤ t ∗with probability 1,• A buyer with v ∈ [kbT , ct∗∗ ) is better off with earlier agreement on a cheaper price pt∗ att ∈ (t ∗ , T ) with probability 1 − α.
• A buyer with v ∈ [c0T , kbT ) is better off with agreement at t = T on a cheaper price p∗T withprobability 1 − α.
• A buyer with v ∈ [cbT , c0T ) is better off with agreement at t = T with probability 1 − α,compared to breakdown with probability 1.
• A buyer with v ∈ [0, cbT ) is indifferent because they cannot trade for both cases withprobability 1.
Overall, this simulation example demonstrates that the soft deadline strategy may be Paretoimproving for every risk-neutral buyer in the demand pool. This assures the participation constraint of each buyer to form a group. In a stark contrast, one can see that a gain for themonopolist (red area) is smaller than the loss (area surrounded by two lines), suggesting thesoft deadline serves as a countermeasure to monopoly power.

Comparative statics of optimal commitment
In Proposition 4, I pin down the optimal commitment to the soft deadline. Next, I assess how
does the commitment policy varies with other primitives, namely, bargaining friction (discountrate) and the market distribution of private values. To see this, given a parameterized modelb with respect to σ and exp(−γ),(T = 6, σ = 1, γ = − ln(0.98)), I simulate the sensitivity of αas depicted in Figure 5.
28 Under the simulation with γ = − ln(0.98),T = 6,t ∗ = 3 at Figure 4, c∗ = 0.60, k∗ = 0.53, c0 = 0.38, c∗ = 0.26
T
T
T
t∗

holds.

20

Figure 5: Optimal commitment under various primitives
Note: A baseline model is simulated with T = 6, σ = 1, γ = − ln(0.98), and a single threat point is set atb with σ > 0 or exp(−γ) > 0,t ∗ = 3. Then, following (7), consider the sensitivity of the optimal commitment α
respectively, all else equal.

b (left) and a higher αb
One can see that σ and exp(−γ) is positively linked with a lower α(right), respectively. To understand the intuition, recall that two forces shaping the response ofmonopoly power a0 from an incremental shift of commitment α (or da0 /dα) are balanced at
b . As as α increases, the exploitive response from strategic interaction surpasses compromiseαresponse from price discrimination (see the decomposition of change in monopoly power inb suggests that price discrimination serves stronger(9) and Figure 3.) Therefore, the higher α
relative to strategic interaction at the soft deadline.
I begin with a straightforward case. As a value distribution parameter σ > 0 increases,29 thedemand pool exhibits more upward-based willingness to pay. As the buyers on average haveless incentive to wait, and the monopolist enjoys a larger benefit from strategic interaction, asd(da0 /dα)b (i.e.;suggested by the lower α> 0). In contrast, as a discount rate γ > 0 decreasesdσ(or equivalently, a periodic discount factor exp(−γ) increases), the bargaining is less frictional
and the monopolist suffers from self-competition by a weaker commitment on pricing. Theb (i.e.;monopolist undergoes a larger loss from price discrimination, as indicated by the higher αd(da0 /dα)< 0). The pair of sensitivity tests on optimal commitment is insightful regardingdγ
the policy implications of soft deadlines.
29 Recall that σ captures the first-order stochastic dominance of the cumulative distribution function of private

value, F(v) = vσ .

21

5Efficiency

In the previous section, I discussed the sensitivity of consumer surplus with the credibility ofa soft deadline. In the durable goods monopolist model, the consumer surplus is often interchangeably associated with the overall market efficiency. I close my analysis by briefly exploring how the bargaining efficiency responds to commitment intensity. I define the bargaining
efficiency U ≡ V0 +W0 as the sum of the players’ ex-ante expected payoffs at t = 0 before theprivate value v is realized. An immediate corollary of Proposition 4 is given as follows.
cd ∈ (0, 1)
Corollary 2. [Efficiency impact of the soft deadline] In the limit as ∆ → 0, α
uniquely maximizes U.
[Proof.] From Corollary 1, V0 =

σ
σ +2
σ
a0 and W0 =(1 −a0 ) holds. Therefore,
σ +1
σ +1
σ +1

U ≡ V0 +W0 =

σ
a0
(1 −
)
σ +1
σ +1

(10)

holds. Because U is strictly decreasing with a0 , the rest follows the proof of Proposition 4. k

As (10) shows, the efficiency U decreases with an ex-ante monopoly power a0 . As theconsumer surplus W0 is negatively associated with the monopoly power a0 (Corollary (1)),
the model links the consumer surplus to the overall efficiency in the monopolistic market.
Therefore, an inverted-U sensitivity of the buyer’s expected surplus at Proposition 4 is inherited
to the sensitivity of efficiency as well.
Mirrored by the discussion on consumer surplus, this result is also intriguing from theperspective of market design of durable good transactions. Under the classical durable goods
monopolist model, a longer trade horizon (or durability) implies larger efficiency. Consider
the two extreme cases again as in section 4. Under the one-shot game (T → 0), the bargaining
undergoes the largest distortion from the strongest monopoly power with an ultimatum. The
bargaining closes instantaneously, but a significant share of buyers cannot buy the goods in theface of the monopoly pricing. In the infinite horizon (T → ∞), however, the bargaining achieves
Pareto efficiency, consistent with the Coase conjecture. All the buyers enjoy a competitive
pricing with no delay. One may observe that the simple perturbation of deadline structureenhances the efficiency through the resurgence of the self-competition dynamics under thedeadline.
22

6Concluding Remarks

The monopolist credibly exploits a hard deadline as a commitment device to create his brinkmanship. This paper explores a new commitment technology by a consortium of buyers—a soft
deadline to counter the monopoly. Using a simple durable goods monopolist model under adeadline, I demonstrate the possibility that the buyers’ imperfect commitment to earlier exit
potentially augments the buyers’ expected surplus by creating a new motive for the monopolist
to engage in price discrimination. One may observe that the soft deadline partially revives the
dynamics of price discrimination which had been dormant under the hard deadline; when thecompromise from price discrimination dominates the exploitation from the strategic interaction
at the soft deadline, the imperfect commitment to a soft deadline serves as a countermeasure tomonopoly power. This finding revisits conventional wisdom regarding the durable goods monopolist model to relate durability to monopoly power, which lies at the heart of the Coasianlogic.
Three caveats are worth noting. First, to adopt a soft deadline strategy, one might consider
the best timing and numbers of soft deadlines, upon which this paper is silent.30 Second, monopolists are plausibly armed with richer outside options than the buyer’s side. Nevertheless,the model presumes the common outside options for both parties. Third, although my model
imposes risk neutrality for both parties, either party might be risk-averse or risk-loving in thereal world, which is potentially associated with their outside options. The distributional implication may depend on the relative strength of outside options and risk preferences. These
points are left for future work.

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Appendix
Multiple soft deadlines
Instead of the simplest figure with a single soft deadline (N = 1), I consider a multiple cases(N > 0).

Figure 6: multiple soft deadlines
Note: A soft deadline with a conditional probability of breakdown αd ∈ (0, 1) is imposed at the end of time td
in contrast to the hard deadline at time T . The buyer’s group commits that the negotiation might end at td withprobability αd but continue with probability 1 − αd .

Discrete period vs. Continuous time
To rigorously illustrate the construction of variables continuous time limit, take the following
steps. First, pick up any t ∈ [0, T ] and let t = m∆, where m is a non-negative integer withTm ∈ {1, 2, · · · , M} and a length of a period ∆ > 0. Second, one can define M ≡ d e, where dxe∆
is the largest integer that satisfies dxe ≤ x. Third, as a period of each bargaining round shrinks(i.e., ∆ → 0), variables are defined in the continuous time limit.

Derive bargaining powers {at } and {bt }To solve the model of (2), (3) and (4), I use backward induction in Sobel and Takahashi(1983).31 As in (5), price and cutoff schedules are respectively characterized by bargaining
powers {at } and {bt }. For analytical convenience, use xt ≡ (bt )−1 (∀ t∈ [0, T ]) instead of bt
31 See the mathematical appendix in Fuchs and Skrzypacz (2013). A detailed proof is available on request.

29

Figure 7: Construction of continuous-time variables
below. Then, I derive at by a difference equation of {at }:

at =

xt
(σ + 1)xt − σ ηt exp(−γ∆)at+∆

1
σ

−1

xt , aT = (1 + σ ) σwhere xt = 1 − ηt exp(−γ∆)(1 − at+∆ ) and xT = 1. (Recall (6)) Then, translating the difference equation into a differential equation and solving it based on Ausubel and Deneckere(1992) yields


∗


at = e−γ(t1 −t) at1
at = e−γ(td+1 −t) atd+1





at = e−γ(T −t) aT(0 ≤ t ≤ t1 )(td < t ≤ td+1 , (d ∈ {1, · · · , N − 1})(tN∗ < t ≤ T ),where
atd = (

1xtd
) σ xtd (d ∈ {1, · · · , N})(1 + σ )xtd − σ (1 − αd )limat

t↓td


=

1


)1+σ σ(xtd
σ αd + xtd={αd + (1 − αd )limat }1+σ

 σ1

(1 + σ )αd + (1 − αd )limat



t↓td

t↓td

30

(11)

holds. In the limit as ∆ → 0, I obtain {bt } as
btd = {1 − (1 − αd )(1 − limat )}−1 = {αd + (1 − αd )limat }−1


t↓td
t↓td


1
bt = {1 − (1 − at )}−1 =

at




b = 1T(td (d = 1, · · · , N))(t ∈ [0,t1 ) ∪ (td ,td+1 ) ∪ (tN∗ , T ))(t = T ).

Proof of Lemma 1 [Deadline effect]
To derive the cutoff drop at td , I characterize kt by solving a differential equation based on (1).
By straightforward algebra,
kt (∆) = ct−∆ (∆) =at−∆ (∆)k(∆)
xt−∆ (∆)=
kt−∆ (∆)
xt−∆ (∆)
2xt−∆ (∆) − e−γ∆ at (∆)

holds. Rearranging it, one getskt (∆) − kt−∆ (∆) −xt−∆ (∆) + e−γ∆ at (∆) kt−∆ (∆)=.
∆∆
2xt−∆ (∆) − e−γ∆ at (∆)
Thus, I obtain a differential equation such thatdktdt

kt (∆) − kt−∆ (∆)−γ∆ kt−∆ (∆)γ= lim= − kt .
∆→0∆→0 at (∆)
∆
∆
at

≡ lim

Dividing into time intervals, kt must satisfy these differential equations.



dkt

γ


= − −γ(t −t) kt


1
dt
e
at1




dkt
γ
= − −γ(t −t)
kt


d+1
dt
e
a

t
d+1




dkt

γ



= − −γ(T −t) kt
dteaT(0 ≤ t ≤ t1 )(td < t ≤ td+1 , d ∈ {1, · · · , N − 1})(tN∗ < t ≤ T )Solving the three equations above given the boundary conditions k0 , limkt (d ∈ {1, · · · , N − 1}),
t↓td

31

limkt , respectively, one gets
t↓tN



eγt1 −γt(e − 1)}(0 ≤ t ≤ t1 )kt = k0 exp{


at1



eγ(td+1 −td ) −γ(t−td )kt = limkt exp{(e− 1)} (td < t ≤ td+1 )

atd+1
t↓td




eγ(T −tN ) −γ(t−tN )


k
=
lim
k
exp{
(e
− 1)}
(tN < t ≤ T ),
 t
t
aT
t↓tN

(12)

where

limkt = ctd =
t↓td

 σ1

αd + (1 − αd )limat



1
xtd
atd
t↓td
 kt
ktd = (
) σ ktd = dxtd
σ αd + xtd(1 + σ )αd + (1 − αd )limat

(13)

t↓td

holds. These imply that in the three domains above, kt is continuous. The cutoff drop g(αd ) at
t = td (d ∈ {1, · · · , N}) is formulated byg(αd ) ≡ lim ct − ctd = (1 −
t↑td

atd
)kt
xtd d
1

αd + (1 − αd )limat= {1 − (t↓td(σ + 1)αd + (1 − αd )limat{(σ + 1)αd + (1 − αd )limat } σ (1 − eγtd )

1
σ

t↓td

) } exp

σ +1.

{αd + (1 − αd )limat } σ

t↓td

t↓td

One can check that g(αd ) ≥ 0 if and only if αd ≥ 0, and g(0) = 0. Note g(αd ) is a function ofprimitives αd , σ , and limat , which is determined by σ , γ, td , and T . Because g(0) = 0, whatt↓td

remains to be shown is for all αd ∈ (0, 1),∂ g(αd )> 0. The first-order condition yields that∂ αd(1 − αd )limat + αddlimktlimat t↓td

t↓td

(1 − αd )limat + αd (σ + 1)

 σ1
 kt
d

d(atd /xtd )dg(αd )
t↓td
t↓td
=−
=−
ktd =
>0
dαd
dαd
dαd
((1 − αd )limat + αd )((1 − αd )limat + αd (σ + 1))
t↓td

t↓td

holds. The first equality holds because ktd is independent from αd . This finding implies thecutoff drop is strictly increasing in αd . k

32

Proof of Proposition 2. [Sales after the threat point]
Plugging (11) and (12) into pt = at kt , I obtain the price schedule:eγ(t1 −t) −γt(e − 1)}e−γ(t1 −t) at1
p
=
k
exp{
t
0


a

t1


eγ(td+1 −t) −γt
pt = limkt exp{(e − 1)}e−γ(td+1 −t) atd+1

a
t↓t
d
t

d+1


γ(T −t)e(e−γ(T −t) − 1)}e−γ(T −t) aT
 pt = limkt exp{aTt↓tN

(0 ≤ t ≤ t1 )(td < t ≤ td+1 (d ∈ {1, · · · , N − 1}))(tN < t ≤ T ).

Recall that the drop of the price schedule h(αd ) is a function of primitives αd , σ , and limat ,t↓tdwhich is determined by σ , γ, td , and T . Specifically,h(αd ) ≡ ptd − lim pt = atd ktd − limat limkt =
t↓td

t↓td

t↓td

atd ktd(1 − limat ) αd
xtd
t↓td
{z
}
|
>0

αd + (1 − αd )limat
=(

t↓td

1

) σ (1 − limat )ktd αd .
αd (σ + 1) + limat (1 − αd )
t↓td
|
{z
}
t↓td
independent of αd

Because h(0) = 0, what remains to be shown is for all αd ∈ (0, 1),dh(αd )
> 0. The first-orderdαdcondition yields
dh(αd )= (1 − limat )(1 + σ )αd2 + (−1 + σ )limat αd + limat .
dαd
t↓td
t↓td
t↓td
The second-order condition yields
d 2 h(αd )= 2(1 − limat )(1 + σ )αd + (−1 + σ ).
t↓td
dαd2
Solvinged )1−σ
d 2 h(αd 2 h(αd )
e
=
0,
one
gets
α
=
such
that
= 0. When σ > 1,d2(1 − limat )(1 + σ )
dαd2
dαd2
t↓td

d 2 h(αd )
dh(0)
dh(0)
> 0 holds. Together with= limat > 0, for all αd ∈ [0, 1],> 0 holds.
2
dαd
dαd
t↓td
dαd
d 2 h(αd )d 2 h(αd )
ed ),
e
<
0,
and
for
α
∈
(
α
,
1),
> 0 holds. Then,
When 0 < σ ≤ 1, for α ∈ [0, α
d
dαd2
dαd2

33

evaluatingdh(αd )
ed , one finds thatat the minimizer αd = α
dαd
ed )
dh(α
ed2 + (−1 + σ )limat α
ed + limat= (1 − limat )(1 + σ )α
dαd
t↓td
t↓td
t↓td
2
(1 − 2limat )(1 − σ )t↓td+ limat > 0=4(1 − limat )(1 + σ ) t↓td
| {z }
t↓td
|
{z
}
>0
>0

−1

holds because when 0 < σ ≤ 1, limat < aT = (1 + σ ) σ <t↓td

1
holds. Therefore, for all αd ∈2dh(αd )
> 0 holds. k(0, 1), and σ > 0,dαdProof of Proposition 3. [Discount before the threat point]
First, see the proof of Proposition 4 below, which shows that atd (αd = 0) < atd (αd = 1) holds
cd ∈ (0, 1) that min(i.e., a shorter bargaining is better-off for a monopolist) and there exists α
imizes atd . Because atd is continuous with respect to αd , by the intermediate value theocd , 1) such that atd (α d ) = atd (αd = 0). Then, for αd ∈ [0, α d ],rem, there must be α d ∈ (αda0
atd (αd = 0) > atd (αd ) holds. As> 0, at (αd = 0) > atd (αd ) for all t ∈ [0,td ]. Moredatd
dlimktdkt (αd )t↓td< 0 in Lemma 1,< 0 holds for all αd ∈ [0, 1). (Recall that
over, as I show
dαd
dαd
dkt
> 0 (t ≤ td ) from (12).) Thus, for α ∈ (0, α], pt (α = 0) > pt (α) for all t ∈ [0,td ]. k
dlimktt↓td

Proof of Proposition 4. [Buyers’ premium from the soft deadline]
What remains to be shown is the following two points.
Monotonicity of a0 w.r.t. atd
I show thatda0
> 0. For any d 6= N, I show a0 is strictly increasing in atd . When d = 1,
datda0 = e−γt1 at1

34

(14)

holds. For all 1 < d ≤ N,
atd = lim (1 − δ + δ Atd +∆ (∆))(∆→01
1 − δ + δ Atd +∆ (∆))σ(σ + 1)(1 − δ ) + δ Atd +∆ (∆)

1+σ{αd + (1 − αd )limat } σ
t↓td

=

1

{(σ + 1)αd + (1 − αd )limat } σt↓tdholds. When αd > 0,1

{αd + (1 − αd )limat } σ

datd
t↓td
=
(15)
σ +1 (1 − αd ){(1 − αd )limat + αd (2 + σ )} > 0dlimatt↓td{(σ + 1)αd + (1 − αd )limat } σ
t↓td

t↓td

holds. For td < t ≤ td+1 (d = 1, · · · , N − 1),limat = e−γ(td+1 −td ) atd+1

(16)

t↓td

holds. Combining (14)∼(16) shows a0 is strictly increasing in atd .
Sensitivity of atd w.r.t. αdRecall that atd is the monopolist’s bargaining power at the threat point td . Using xt instead ofbt , (9) is reformulated asdatd (xtd , αd ) datd dxtd ∂ atd
=
+
= 0.
dαd
dxtd dαd ∂ αd(17)

By some algebra, the monopolist’s strategic interaction is captured bydatdxtd= (1 + σ )αd1+σ (> 0)
dxtd{limat + αd (1 + σ − limat )} σt↓td

and the buyer’s response is captured by

t↓td

dxtd
= 1 − limat (> 0). By contrast, the monopolist’s
dαd
t↓td

compromise is given by
∂ atd
xtd
= limat {limat (αd − 1) − αd }1+σ (> 0).
∂ αd t↓td
t↓td{limat + αd (1 + σ − limat )} σ
t↓td

35

t↓td

Combining these, (17) is reduced to(limat )2 (αd − 1) + αd {1 + σ − limat (2 + σ )} = 0.
t↓td

t↓td

Solving for αd , one gets
(limat )2
cd = arg min atd =αt↓td(1 − limat )(1 + σ − limat )
t↓td

s.t.

t↓td

cd )
∂ atd (xtd , α
= 0.
∂ αd

(18)

The second-order condition yields
1−σ
d 2 atd= (limat )2 (1 + σ ){(1 − αd )limat + αd } σ
2
t↓td
t↓td
dαd

{(1 − αd )limat + αd (1 + σ )}

−1−2σ
σ

t↓td

(19)

> 0.

(20)

cd is interior such that α
cd ∈ (0, 1). Suppose α
cd ≥ 1; then, by (19),I finally show α1+σ2+σlimat ≥
t↓td

must hold. Because
1

limat < AT = (1 + σ )− σ <
t↓td

1+σ
2+σ

(∀ σ > 0)

cd < 1 holds. Finally, because limat ∈ (0, 1) holds,
holds, this is a contradiction. Therefore, α
t↓td

cd =
α

(limat )2t↓td(1 − limat )(1 + σ − limat )t↓td> 0 holds. From (18) and (19), atd is positively quadratict↓td

cd ∈ (0, 1). By (8), W0 is negatively quadratic and maximized with α
cd ∈and minimized with α(0, 1). k

Illustration of strategies of both parties under the optimal commitment
Combining Lemma 1 and Proposition 2, 3,4, I illustrate the cutoff {ct } and price {pt } whenb in Figure 8.
α = 0 vs. α = α36Figure 8: Bargaining dynamics under the optimal commitment
Note: The bargaining might end at åt ∗ = 3 with probability α, but continue with probability 1 − α. The schedulesof the players are simulated, where a soft deadline with the optimal commitment intensity α = 0.274 is imposed
at t ∗ = 3 out of the time horizon T = 6. In equilibrium, the cutoff schedule specifies the timing of purchase for agroup of buyers of different values. See (5) and (6) for characterization of both schedules.

37