Consumer Private Experimentation

Francesc Dilmé∗

June 1, 2015

Abstract

This paper investigates the effects of private experimentation on trade out-comes. We consider a consumer who can costly and privately experiment to knowabout his true valuation of a good. After such learning takes place, a seller makeshim a price offer. In the unique equilibrium outcome, the consumer undertakesexperimentation in order to obtain information rents. The seller chooses her priceoffer using a continuous distribution, and there is a positive probability that notransaction takes place even when the gains from trade are large.

In the limit where the experimentation cost tends to 0, the pricing strategy ofthe seller tends to a degenerated distribution, which lowers the value of experi-mentation. As a result, the amount of information that the consumer chooses toobtain remains bounded away from full information. When the gains from tradeare low, the consumer is better-off when experimentation costs are neither too lowor too high. When, instead, the gains from trade are high, the consumer preferslow experimentation costs, since his endogenous lack of information lowers theaggressiveness of the seller.

Keywords: Private Experimentation, Pre-trade Learning, Price Dispersion.

JEL Classifications: D82, D83, D42, L15

∗University of Bonn. fdilme@uni-bonn.de

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1 Introduction

Before a buyer purchases a good, he can typically gather information about it. Suchinformation may be useful to get a better idea on how valuable is the good to him andto obtain some information rents. As the process of gathering information is typicallyunobserved by the seller of the good, she needs to form beliefs about the informationheld by the buyer when she decides the price. If the seller has monopolistic power, shemay use it in order to lower the information rents of the seller. In such environment,the interaction between the optimal experimentation amount and pricing strategy isfar from trivial: while the buyer enjoys endogenous private information, the seller hasmonopolistic power. The goal of this paper is to analyze how private experimentationshapes the trade outcomes in such environment.

We consider a seller (she) who owns a good. A potential buyer (he) is not sure abouthis true valuation for the good, which can be either low or high. In the first stage,the buyer can privately undertake costly experimentation. Experimentation impliesincurring a flow cost in order to observe an informative stochastic process (which takesthe form of a standard diffusion), interpreted as a sequence of lowly-informative signals.Once the buyer stops experimenting, the second stage begins. The buyer meets a sellerwho does not observe the first stage. The seller makes a take-it-or-leave-it-offer to thebuyer, who finally accepts or rejects it.

We show that the buyer undertakes some experimentation in all equilibria. Eventhough experimentation does not increase his expected valuation, he has the option ofonly purchasing the good if he obtains signals that indicate that his true valuation forthe good is high. This allows him to extract information rents from the seller. Onthe contrary, if there was no equilibrium experimentation, the seller would offer a priceequal to the expected valuation of the buyer, giving him no information rents. We findthat the seller is better off when obtaining information is more expensive.

The model features a unique equilibrium trade outcome, defined as the equilibriumdistribution of prices offered by the seller and conditional acceptance probabilities ofthe buyer. The support of the price distribution coincides with the support of the dis-tribution of the buyer’s valuations after experimentation. The inclusion of the supportof prices in the support of valuations is intuitive: a price offer out of the support ofvaluations can be slightly increased and give more revenue to the seller. The reverseinclusion stems from the fact that the buyer should be willing to keep experimentingin the boundaries of the support of valuations, which requires the continuation valueto be strictly convex. Then, the support of the price distribution cannot contain holes,since the continuation value for the buyer would be linear in them.

In the limit where experimentation becomes arbitrarily cheap the pricing policy of

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the seller becomes extreme (either very high or very low prices), which induces thebuyer chooses to remain partially uninformed. When the gains from trade are large,the seller posts prices that are close to the the lowest valuation of the buyer in order toincrease the probability of transaction. In this case, the buyer obtains all informationrents, and trade happens with probability one. When gains from trade are moderate,the lack of information acquisition by the buyer makes the seller less aggressive thanshe would be if the buyer was perfectly informed about his valuation. As a result, suchendogenous lack of information is beneficial for the buyer. When, instead, the gainsfrom trade are low, the seller offers prices which are close to the highest valuation ofthe buyer. Consequently, the information rents obtained by the buyer are small, whichdiscourages information acquisition. In this case the price distribution converges tothat of the model with a fully informed buyer, but the probability of trade remainsnoticeably lower. The buyer is found to be better-off when the cost of experimenting ismoderate instead of when it is either very low or very high.

We extend our analysis to allow for a general stochastic process. In this context,we reverse the roles of the buyer and the seller, that is, we consider a setting where an(endogenously) informed seller meets an uninformed buyer. The seller can undertakea sequence of small unobservable investments in order to stochastically increase thequality of the good that is traded. Then, the buyer makes a take-it-or-leave-it offerto the seller. We show that when investing is efficient for the buyer but not for theseller, uncertainty about the outcome of the investments alleviates the implied hold-upproblem. In this setting, we also study competition among buyers. When two or morebuyers Bertrand-compete in order to acquire the product, the price distribution eithercontains a single price or has an interval and an isolated point as support.

The organization of the paper is as follows. After this introduction, we review therelated literature. Section 2 presents our base model. In Section 3 we extend ourprevious analysis to the case of private investments by the seller. Section 4 concludes.An appendix contains the proofs of all lemmas and propositions of the previous sections.

1.1 Related Literature

Our model contributes to the literature on private information gathering before trade.Crémer, Khalil and Rochet (1998a) study strategic information gathering before a con-tract is offered. They find that the agent may randomize between acquiring (full)information or not. Bar-Isaac, Caruana and Cuñat (2012) study information-gatheringexternalities in a model where consumers choose to obtain information on one of theattributes of an exogenously-priced good after its producer privately makes a privateinvestment. They find that lowering the information-gathering cost of one of the at-

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tributes may reduce equilibrium investment of the seller in the other attribute, andlower the equilibrium surplus of the consumer. Similarly to them, we find that the in-formation that our buyer gathers is bounded away from full information, and our sellermay price more aggressively when information becomes cheaper.

Some papers in the literature analyze the case where the principal observes theamount (or precision) of the information gathering, but not its outcome. Kessler (1998)analyzes a model an agent chooses (at a cost) the precision of a signal before beingoffered a contract by a principal, who observes the precision choice but not the signal.She finds that even when the cost of gathering information is low, it is optimal forthe agent to not acquire a signal that is too precise.1 Alternatively, Bergemann andPesendorfer (2007) and Ganuza and Penalva (2010) consider an auctioneer who decideshow much information to provide to the bidders. We ask similar questions in a modelwhere the buyer undertakes sequential experimentation. The fundamental difference isthat we drop the assumption of observability by the principal (seller), who is totallyuninformed about the experimentation choice by the agent (buyer).

Ours is an experimentation model where we endogenize the value of stopping ex-perimentating through trade stage. Other papers in the experimentation literature usediffusions (see, for example, Karatzas (1984) and Moscarini and Smith (2001)), butassume a fixed value of stopping experimenting. Our main focus is on the fix pointproblem between the incentive of experimenting by the buyer and the amount of ex-perimentation believed by the seller.

Finally, our paper also relates to the literature on mechanisms with endogenous in-formation. Among others, Crémer, Khalil and Rochet (1998b), Persico (2000), Berge-mann and Välimäki (2002), Shi (2012), Yang (2013) and Terstiege (2015) consider thedesign of contracts/auctions (or other mechanisms) where agents/bidders can privatelyacquire information. We relax the assumption of commitment by the principal, that is,we only let the principal choose the price after experimentation has taken place. Then,we focus on solving the fixed point between the buyer optimization on the amount ex-perimentation given the anticipated price distribution, and the seller’s pricing decisionin response to the believed amount of experimentation.

1Also in the Bayesian persuasion literature, initiated by Gentzkow and Kamenica (2011), an agentchooses the information structure in order to influence the behavior of other agents. The key questionin this literature is which information the agent should (commit to) obtain. Apart from dropping theassumption that the information choice is observable, our main goal is analyzing how much costlyinformation is obtained in equilibrium. The closest paper to ours in this literature is Roesler (2015),and we will relate our findings to her results on the value of ignorance in Section 2.2.

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2 Base Model

There are a seller (she) and a potential buyer (he). The seller owns a good. The sellervalues the good at −g ∈ R,2 while the true valuation of the buyer can be either low,vL, or high, vH , with vH > max0, vL. The payoff for the buyer of not obtainingthe object is 0. Initially, the buyer and the seller are equally informed about the truevaluation of the buyer. They share a common prior φ0 ∈ (0, 1) about the true valuationof the buyer being high. We assume that there are expected gains form trade, that is,−g < v0 ≡ φ0 vH + (1− φ0) vL (this assumption is relaxed in Section 2.3). We assumethat both the buyer and the seller are expected utility maximizers and risk-neutral.

The game is divided in two stages. In the first stage, the buyer can experiment inorder to obtain some information about his true valuation. The first stage is unobservedby the seller. In the second stage, the seller makes a take-it-or-leave-it offer to thebuyer.3 The buyer accepts or rejects it and the game ends.

Experimentation Stage

The buyer can experiment in the first stage of the game. We model experimentationas the opportunity of exerting effort in continuous time in order to obtain a signal thatis informative about his valuation. Obtaining the signal costs c > 0 per unit of time.Once the buyer stops experimenting the first stage ends.

As is standard in the experimentation literature with diffusions, we assume thebuyer observes the value of a diffusion process (Xt)t. The drift of this diffusion is µθif his true valuation for the good is vθ ∈ vL, vH, with µH > µL, and the variance isσ > 0 independently of the true valuation. So, the stochastic equation followed by Xt

when the true valuation is vθ is

dXt = µθ dt+ σ dWt ,

where Wt is a Wiener process on a probability space (Ω,F ,P). In this case, usingthe continuous-time Bayes’ rule, the (expected) valuation of the buyer v is a diffusion

2Even though the assumption of (potentially) negative valuation of the good by the seller may seemunnatural (typically one assumes free-disposal), it is just a notationally-convenient normalization.Since the gains from trade are vθ + g if the true valuation is vθ ∈ vL, vH, g is a measure of thegains from trade. Section 2.3 allows for a production cost for the seller and Section 3 relaxes theassumption that the seller’s valuation of the object is independent of the valuation of the buyer.

3Riley and Zeckhauser (1983) show that a take-it-or-leave-it offer is the seller-preferred mechanism.

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process with drift and volatility given by, respectively,4

µ(v) = 0 and σ(v) =(µH − µL) (vH − v) (v − vL)

(vH − vL) σ. (2.1)

Strategies

A strategy of the seller is a stopping time rule τ and an acceptance decision α. Asusual, τ is a random variable from Ω to R+ ∪ ∞ that is Ftt-measurable, whereFt ≡ σ(Xs : s < t) is the sigma algebra induced by the possible observations of theprocess Xs before time t. The acceptance decision α is a function from the outcomesof the first sage (that is, ∪∞t=0Ft) and the price offer received from the seller (in R) to aprobability of accepting the offer. A strategy by the buyer is a distribution over priceoffers Fp ∈ ∆(R), where Fp is interpreted as the CDF of the prices offered.

2.1 Equilibrium Analysis

We call an equilibrium a perfect Bayesian equilibrium of our game. We use Fv to denotethe CDF of the values that the stopping time rule establishes. In order to focus on theeconomically-relevant features of our equilibria, we call a (equilibrium) trade outcomea pair (Fv, Fp) (derived from an equilibrium).5

Let’s begin with a simple but useful preliminary result. The following lemma es-tablishes that, given τ and Fp, there is an essentially unique equilibrium acceptancedecision:

Lemma 2.1. Let (τ, α, Fp) be an equilibrium, and let α be an acceptance decision definedby α = 1 if vτ ≥ p, and 0 otherwise. Then, (τ, α, Fp) is also an equilibrium, andgenerates the same distribution over trade outcomes.

Lemma 2.1 is a standard result for take-it-or-leave-it offers. If there is a mass pointat some v (i.e., if Pr(vτ = v) > 0) then the equilibrium probability of acceptance of anoffer p = v is necessarily equal to 1, since otherwise the seller has the incentive of offerslightly less in order to increase significantly the acceptance probability at an arbitrarilysmall cost.

4If φt is the posterior about the true valuation of the buyer is being high, his expected valuation equalsthe value is vt ≡ φt vH + (1− φt) vL.

5In the introduction we refer to (equilibrium) trade outcome as an “equilibrium distribution of pricesoffered by the seller and conditional acceptance probabilities of the buyer”. Still, given Lemma 2.1,it is easy to verify that there is a one-to-one relationship between such concept of trade outcome andthe one used in the main text.

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Beliefs and Price Supports

Let’s fix an equilibrium (τ, α, Fp). Let V denote its support of Fv. The following lemmaestablishes that there is no equilibrium without experimentation:

Lemma 2.2. There is no equilibrium where |V| = 1.

The intuition behind Lemma 2.2 is the following. Assume that there is an equi-librium where |V| = 1. Since vt is a martingale, this is only possible if there is noexperimentation, so V = v0. Then, given Lemma 2.1, it is optimal for the seller tooffer a price equal to v0, which is accepted for sure. The equilibrium payoff for the buyerin this case 0. Assume that the buyer deviates and undertakes some experimentation.If his valuation reaches some value v ∈ (vL, vH), his payoff in the second stage is 0 ifv < v0 (in this case he does not accept the price offer) or v − v0 if v ≥ v0 (in this casehe accepts the price offer). Given that this payoff function has a kink at v0 (in Fig-ure 1 it corresponds to the kink between 45 dashed line passing through (E[p], 0) andthe horizontal axis, replacing E[p] by v0), standard optimal stopping analysis revealsthat it is optimal for the buyer to undertake some experimentation, which leads to acontradiction.

We use V (vt) to denote the expected continuation value of the buyer at time t(which only depends on vt). Also, we define v− ≡ inf V and v+ ≡ inf V . Since byLemma 2.2 there is no equilibrium without experimentation and vt is a martingale,we have v− < v0 < v+. Since vt is a continuous process and the buyer is willing toexperiment until vt reaches v− or v+, using standard stochastic calculus we obtain thatV (·) is differentiable in [vL, vH ] and twice differentiable in (v−, v+), where satisfies thefollowing Hamilton-Jacobi-Bellman (HJB) equation:

0 = −c+1

2σ(v)2 V ′′(v) . (2.2)

Let P denote the support of Fp. The following proposition establishes that P coin-cides with V , and they take the form of an interval:

Proposition 2.1. The equilibrium supports of prices and expected valuations coincideand have no gaps, P = V = [v−, v+].

The right inclusion of the previous result (that is, P ⊂ V) is intuitive. Indeed,any price that the seller is willing to offer should leave at least one of the “types”(interpreting the expected valuation of the buyer as his type) with 0 information rent.Otherwise, such price could be slightly increased with the same acceptance probability,inducing a profitable deviation. More formally, assume p ∈ P but p /∈ V . If p > v+

the offer is accepted with probability 0, which is dominated by offering p = v0 (since,

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by the martingale property of the expected continuation value, Pr(vτ > v0) > 0). If,instead, p < v+, then [p, p + ε] 6⊂ V for some ε > 0 (since V is a closed set). Hence,p+ ε is accepted with the same probability as p, but generates more revenue.

The left inclusion (that is, P ⊃ V) is less obvious, and it is related with the absenceof gaps in the supports of the distributions. Assume that (p1, p2)∩P = ∅ but p1, p2 ∈ P .Then

V (p2) =

∫ p2

v−

(p2 − p) dFp(p) = V (p1) + Pr(p ≤ p1) (p2 − p1) . (2.3)

If the expected valuation of the buyer is v ∈ (p1, p2), he has the option of stoppingexperimenting. Since he has to be (weakly) willing not to stop, we have

V (v) ≥∫ v

v−

(v − p) dFp(p) = V (p1) +V (p2)− V (p1)

p2 − p1

(v − p1) .

This is a clear violation of the strict convexity of V implied by equation (2.2). Ifv < inf P or v > supP a similar argument can be given using the intervals (v−, inf P)

and (supP , v+), respectively.

Unique Equilibrium Trade Outcome

Using standard optimal-stopping analysis, an immediate but important consequence ofProposition 2.1 is the characterization of the smooth pasting conditions for V :

Corollary 2.1. For any equilibrium strategy, v− and v+ are such that V satisfies:

boundary conditions: V (v−) = 0 and V (v+) = v+ − E[p],

smooth-pasting conditions: V ′(v−) = 0, and V ′(v+) = 1.

Figure 1 plots a particular solution of (2.2) which satisfies the smooth-pasting condi-tions, for some fixed E[p]. When the buyer’s expected valuation reaches v+ he buys thegood for sure, so he obtains v+−E[p]. If his expected valuation reaches v−, instead, hestops experimenting, and since he is offered a price equal or higher than v−, he obtainsno information rent. The horizontal axis and the dashed line, indicate, respectively, thevalue of not purchasing the good or purchasing the good for sure.

The following proposition establishes the uniqueness of the equilibrium trade out-come, that is, all equilibria have the same value and price distributions.

Proposition 2.2. Any equilibrium has the same trade outcome, (Fv, Fp), with intervalsupport, [v−, v+], satisfying:

1. Fv(v) = v−v−v+g

for v ∈ [v−, v+) and Fv(v+) = 1,

2. F ′p(p) = 2 cσ(p)2

for p ∈ [v−, v+].

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v

V

v+−E[p]

v− E[p] v+

Figure 1: Equilibrium continuation value function.

Fp has no mass points, while the only mass point of Fv is v+.

The absence of mass points of Fv except for v+ is implied by the indifference of theseller on offering any price in V . If Fv has a mass point at some v < v+, offering a pricep = v + ε is dominated by offering p = v, for ε > 0 small enough. Indeed, even thoughoffering p = v slightly lowers the revenue per unit of probability of trade, it generates anot-small increase the probability of trade. This argument does not apply at v+, wherethere must be a mass point in order for the seller’s payoff to be positive when offeringa price equal to v+. In order to get an intuition for the absence of mass points in Fp,consider the first equality in the equation (2.3) (which applies to any p2 ∈ P). A masspoint in Fp generates a kink in V ′, which is inconsistent with the differentiability of thevalue function.

The probability density function (PDF) of the price distribution is leaning towardsp = vH+vL

2. Indeed, part 2 of Proposition 2.2 establishes that F ′p(p) is decreasing if

p < vH+vL2

and increasing if p > vH+vL2

(recall equation (2.1)). To see why, assume thatthe expected valuation at time t is vt and the buyer is considering to experiment untils > t and then stopping. The expected continuation value at time s of experimentingis given by

E[V (vs)] =

∫ vH

vL

∫ vs

vL

(vs − p) dFp(p) dF (vs)

=

∫ vt

vL

(vt − p) dFp(p)︸ ︷︷ ︸=V (vt)

+

∫ vH

vL

∫ vs

vt

(vs − p) dFp(p) dF (vs)︸ ︷︷ ︸≡(∗)

,

where F is the CDF of vs conditional on vt. Note that the term inside the integral of(*) is positive for all vs. This term incorporates the extra information rents that theadditional experimentation gives to the buyer: he only buys the good if the price islower than his valuation. Let’s assume s is close to t, and let’s approximate the PDF

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of prices around vt by F ′p(vt) (higher order terms on its Taylor expansion are irrelevantwhen s t). In this case,

E[V (vs)] = V (vt) + 12σ(vs)

2 F ′p(vt) (s− t) + o(s− t) .

As we see, the information rents depend on the “density” of prices at vt, since the extrainformation is useful only if it potentially changes the decision of the buyer. Also,information rents depend positively on the volatility of the expected valuation, sinceit is a measure of the informativeness of the signal.6 Since in equilibrium the buyer isindifferent on keeping experimenting, the information rents have to equal the cost ofexperimenting (which is equal to (s− t) c). Then, F ′p(v) is small when Bayes updatingis fast (so σ(v) is large), reaching its minimum at v = vH+vL

2.

2.2 Comparative Statics

Big Gains from Trade

Lemma 2.2 and Proposition 2.1 establish that, in (any) equilibrium, there is a positiveprobability that the price exceeds the valuation of the buyer. In effect, even thoughthere is common knowledge of gains from trade, these gains from trade may not bematerialized. This feature is present in a variety of models, where a monopolist tradesoff the quantity traded and the price charged.

Consider, for example, the case where the buyer is (exogenously) fully informedabout his true valuation (discussed briefly below). In this case, when the gains fromtrade are large (i.e., when g high enough), the seller charges a price equal to vL in orderto ensure trade. It is then worth to see how our equilibrium trade outcome changes asg increases. The following proposition describes the corresponding comparative statics:

Proposition 2.3. As g increases, both v− and v+ decrease. In the limit where g →∞,we have v+ → v0 and Fv tends to a distribution degenerated at v0.

A direct implication of Proposition 2.3 is that, as g increases, the probability oftrade converges to 1. Even though the support of the distribution of valuations of thebuyer remains big, most its probability mass is at v+, so the price is almost-surelylower than the buyer’s valuation. Even though the seller is “almost convinced” that thevaluation of the buyer is v+ ' v0, she randomizes over a relatively extensive range ofprices, with a distribution which is skewed towards low prices. Even though increasinga price does not decrease the probability of trade by much, the potential loss in gainsfrom trade is big.

6Even though it may be counterintuitive, a quick look at equation (2.1) reveals that σ(v) is increasingin the in µH−µV

σ , which is a measure of how informative is the experimentation.

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The ex-ante payoff of the buyer, given by V (v0), is increasing in g (see the proofof Proposition 2.3). Intuitively, as g increases, the buyer becomes less aggressive inorder to ensure trade. As g → ∞ we have that V (v0) → v0, so the buyer obtains allinformation rents in the limit.

High Cost of Experimentation

Let’s now focus on the effect of rising the cost of experimentation c. The followingproposition establishes that increasing the cost of experimentation decreases the amountof equilibrium experimentation. Since the payoff of the seller is equal to v−, the selleris always better off when the cost of experimentation increases.

Proposition 2.4. v− is increasing in c and v+ is decreasing in c. When c→∞, bothtend to v0, and the probability of trade converges to 1.

As c → ∞, the trade outcome of our model converges to the case where gatheringinformation is impossible. Indeed, if information cannot be gathered, the seller knowsthat the valuation of the buyer is v0, so in the unique equilibrium she offers p = v0 andthe buyer accepts for sure.

Almost-Costless Experimentation

If private learning about a payoff-relevant state becomes cheaper, one could expectthat the buyer will tend to gather larger amounts of information. In our model, suchincrease in the information acquisition should be taken in account by the seller whendetermining her pricing strategy. In particular, if the seller anticipates that a lot ofinformation is obtained by the buyer, she may become more aggressive and offer a highprice in order to sell to the buyer only if his valuation is high. This may lower theincentive to undertake experimentation in the first place. In this section we analyzethe interaction of these two countervailing forces in the limit where experimentationbecomes arbitrarily cheap.

In order to better understand our findings below, we compare them to the analo-gous results in a model where the buyer is (exogenously) perfectly informed about hisvaluation of the good. Assume then that it is common knowledge that the buyer knowshis true valuation (equal to vH with probability φ0 and to vL with probability 1− φ0).In this case, it is easy to verify that, in the only equilibrium, the seller charges vH ifg < g′ and vL if g > g′, where g′ ≡ φ0 vH−vL

1−φ0 . Intuitively, when there are low gains fromtrade, the seller prices the good aggressively and charges vH in order to attract onlythe buyer with high valuation, so trade happens with probability φ0. As the gains fromtrade increase, the seller is more willing to establish a low price in order to attract thebuyer independently of his valuation.

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g

v∗−, v∗+

v0

g

v∗−

−v0

1

v∗+

0vL

Figure 2: v∗− and v∗+ as a function of g.

We are now interested in a small but positive cost of experimenting.7 The follow-ing proposition establishes that, generically, the information that the buyer gathers inequilibrium does not converge to full information when c→ 0.

Proposition 2.5. There exists a threshold g < g′ (which solves (ev0/g − 1) g = 1) suchthat, as c gets close to 0:

1. If g < g then v+ approaches vH and v− approaches v∗−>vL, which is the solutionof v0 = v∗−+(v∗−+g) log

(g+vHg+v∗−

). The price distribution converges to a distribution

degenerated at p=vH and the probability of trade converges to g+v−g+vH

<φ0.

2. If g > g then v− approaches vL and v+ approaches v∗+ ≡ (ev0/g − 1) g < vH .The price distribution converges to a distribution degenerated at p = vL and theprobability of trade converges to 1.

Figure 2 illustrates the implications that Proposition 2.5 has for the support ofthe equilibrium valuations of the seller. Even when the cost of acquiring informationbecomes very small, the amount of information that the buyer acquires is bounded awayfrom full information.

If the gains from trade are low, g < g, the pricing strategy of the seller coincideswith the one of the full-information model: as c → 0, the seller becomes increasinglyaggressive, so the price distribution approaches a distribution degenerated towards vH .The aggressiveness of the seller lowers the incentive to experiment of the buyer, so thebuyer stops experimenting when vt reaches v∗− > vL. This implies that the limit equilib-rium trade outcome of our model does not coincide with the equilibrium trade outcomeof the model with a perfectly informed buyer: even though the price distributions co-

7The limit when c goes to 0 can be analogously interpreted as the signal becoming more informative(so the ratio µ/σ increases).

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g

v0

g−v0

vH

g′0−vH

Payoff of the seller:full informationno information (c→∞)almost-costless info (c→ 0)

Figure 3: Payoff of the seller under full information (black line), no information (dashed line)and almost-costless information (gray line), for vL = 0.

incide, the probability of trade remains bounded away from φ0 in our model, which isthe probability of trade in the full-information model.

When the gains from trade are intermediate, g ∈ (g, g′), our seller’s aggressivenessdecreases as c→ 0, and charges vL in the limit. This is not the case in the full informa-tion model, where the seller finds optimal to charge vH . The reason for such differenceis the lack of information obtained by the buyer: it lowers the payoff for the seller ofposting a high price. Given that, by Proposition 2.2, the density of the price distribu-tion for each v in the support of expected valuations becomes small (and the supportbecomes bigger), the equilibrium prescribes that the seller seta a price which is veryclose to vL with a very high probability. This discourages experimentation, so the buyerstops experimenting when vt reaches v∗+ < vH , which reinforces the low aggressivenessof the seller. As a result, the equilibrium trade outcome of our limiting model differsfrom the full-information model in two dimensions: the price is degenerated towards vLinstead of vH and the probability of trade is 1 instead of φ0.

Finally, when the gains from trade are high, the limiting trade outcome of our modelcoincides with the one of the full information model: the seller charges a price equal tovL and trade happens with probability one. In this case, the possibility of realizing biggains from trade lowers the seller’s aggressiveness.

Figure 3 plots the payoffs of the seller under the full-information, no-informationand almost-costless (c → 0) information models, obtained using the previous results.When g < −v0 there is no trade in the no-information and almost-costless models (seeProposition 2.6 below), so the seller obtains −g. The payoff of the seller when c isneither close to 0 or very large is always contained between the gray curve and blackcurve.

Remark 2.1. If g < g, the payoff for the buyer tends 0 both in the limit c → 0 (sincethe seller becomes aggressive and charges vH) and in the limit c→∞ (since obtaining

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private information is too expensive), while it is positive for intermediate values of c. Inthis case, information needs to be not too costly and not too cheap in order for the buyerto obtain some information rents. This relates to the idea that having (the opportunityto gather) too much information may be bad for an agent. A similar conclusion isobtained in models with observable information gathering, such as Kessler (1998) andRoesler (2015), where they also find that ignorance may be beneficial for the buyer.

When gains from trade are intermediate, g ∈ (g′, g), we also find that remainingpartially informed is good for the buyer. Since even when the cost of experimentationis very small the buyer remains partially uninformed, the seller is less aggressive thatshe would be under perfect information, so she offers a price equal to vL instead of vH .

2.3 Relaxation of Assumptions

Non-Expected Gains from Trade

Let’s first relax our assumption that there are expected gains from trade. In this case,no experimentation takes place and, if −g < v0, there is no trade in equilibrium.

Proposition 2.6. If −g ≥ v0 then the only equilibrium involves no experimentation.

The intuition of the previous proposition is as follows. Assume that the buyerundertakes some experimentation in equilibrium. Since the seller never charges a pricelower than −g (she prefers to keep the good instead), experimentation is optimal onlyif v+ > −g. Let p∗ ≡ infp ∈ P|p ≥ −g > −g. For the same reasons as before, P ⊂ V ,so the buyer is willing to stop experimenting when vt = p∗ and therefore obtain a payoffequal to 0. Finally, since his payoff is also 0 when he does not experiment, it is clearthat experimenting at time 0 is costly and does not increase his expected payoff.

Production Cost by the Seller

In our base model we assume that the seller derives a payoff from retaining the objectequal to −g. This interpretation is given by the assumption that the good is producedbefore the buyer and the seller meet, so they bargain over an existing good. Examplesof such goods are durable goods such as houses, cars, etc.

In some situations, goods are produced after the agreement to trade takes place.Examples are services, experience goods, etc.8 So, we assume that the seller incurs aproduction cost C if trade happens. It is easy to verify that the results of Sections 2.1and 2.2 are still valid if g is replaced with g − C. Intuitively, the gains from trade (−g

8Here we abstract from the moral hazard problems that trade such goods or services involves. Section3 accommodates the case where the production cost is correlated with the valuation of the buyer.

14

is interpreted as the payoff of not producing the good) are reduced because of the costof producing it.

3 Seller’s Private Investments

As in many other trade models, we can switch the roles of the trading parts in our basemodel. This corresponds to considering a situation where an (endogenously) informedseller meets an uninformed buyer. In this context, experimentation can be reinterpretedas investment, which motivates the use of a more general stochastic process. In thissection we generalize many of our previous results, analyze the competitive market andprovide insights on the hold-up problem.

Consider a seller who owns a durable good. She has some initial value for thegood v0 ∈ R (interpreted as quality). She may invest in improving the quality of thegood (which equals the valuation of the seller). Investment takes part over time and isstochastic. The value follows a general diffusion process:

dvt = µ(vt) dt+ σ(vt) dWt

where, as before,Wt is a Wiener process on a probability space (Ω,F ,P), and σ(v) > 0.The flow cost of investing is denoted, with some abuse of notation, c(v), and we assumethat is bounded below away from 0. We assume that µ, σ and c are twice differentiable.

Once the seller stops investing, she meets a buyer. The buyer does not observe thequality resulting from investment, and he values a good of quality v at u(v), whereu(v) > v for all v < v0. He obtains a payoff of 0 if he does not purchase the good. Thebuyer makes a take-it-or-leave-it offer to the seller. If the seller accepts the offer, thegood is transacted, and otherwise the seller keeps the good.

Assumption 1. Investing is inefficient for the seller, that is, µ(v) < c(v) ∀v ∈ R.

Assumption 1 establishes that the expected increase in seller’s value from investingis lower than its cost. This implies that, in the absence of selling considerations, shewould not invest on improving the quality of the good.

Remark 3.1. The investment model is a generalization of the experimentation model inSection 2. In order to illustrate the connection, fix an equilibrium trade outcome of theexperimentation (base) model with initial valuation v0, denoted (FE

v , FEp ). Consider the

investment model with c(v) = c > 0, u(v) = g− vH − vL, and µ(v) and σ(v) as in (2.1)(note that they satisfy Assumption 1). Then, it is easy to show that (F I

v , FIp) with initial

valuation vH +vL−v0 defined by F Iv(v) ≡ FE

v (vH +vL−v) and F Ip(v) ≡ FE

p (vH +vL−v)

for all v ∈ [0, 1], is an equilibrium trade outcome in the investment model.

15

Lemmas 2.1 and 2.2 and Proposition 2.1 still hold in this setting. Now, the HJBequation (2.2) is replaced by a more general one:

0 = −c(v) + µ(v)V ′(v) + 12σ(v)2 V ′′(v) . (3.1)

Also, the boundary and smooth-pasting conditions of Corollary 2.1 now are given by:

boundary conditions: V (v−) = E[p] and V (v+) = v+,

smooth-pasting conditions: V ′(v−) = 0, and V ′(v+) = 1.

The only difference in the boundary conditions between the two settings is given bythe position of E[p] in the expressions. In the experimentation model the buyer paysE[p] when his valuation reaches his highest equilibrium valuation (equal to v+), whileobtains 0 information rents when his valuations reaches v−. In the investment setting,the seller receives E[p] when her valuation is her lowest equilibrium valuation (equal tov−), while she obtains v+ when her valuation reaches the highest equilibrium value.

Proposition 3.1. Assume (Fv, Fp) is an equilibrium trade outcome. Then, it has aninterval support [v−, v+] 3 v0, and satisfies:

1. Fv solves Fv(p) = F ′v(p) (u(p)− p) with Fv(v+) = 1.

2. Fp solves (3.1) replacing V ′(v) by Fp(v) and V ′′(v) by F ′p(v).

Fp has no mass points, while the only mass point of Fv is v−.

Part 1 of Proposition 3.1 provides us with an intuition of the trade-off that the buyermakes at the margin. Consider a buyer that considers increasing the price offer from p

to p+ ε, for some ε > 0 small. As usual, this increases the payment to the seller’s typesthat sell the good when the price is p. The corresponding increase in this payment isequal to Fv(p) ε. Also, since the price is higher, there are marginal types of the sellerthat now agree to sell. Since the marginal type has valuation equal to p, the incrementin the gains from trade is F ′v(p) (u(p)− p) ε+ O(ε2). Therefore, the buyer can only beindifferent between offering p and p+ ε if Fv(p) = F ′v(p) (u(p)− p).

Another implication of Part 1 of Proposition 3.1 is that the seller stops investingwhen her expected valuation is v only if there are gains from trade at v, that is, ifu(v) > v. As a result, conditional on the good being transacted, there is no ex-postregret (or re-trade): u(v) ≥ v for all v ∈ V .

3.1 Hold-Up-Problem Limit

There is an extensive literature on the “hold-up problem”, that is, the underinvestment(or not investment at all) by a seller due to the inability of the buyer to commit to a

16

price offer. In our model the fact that investment is stochastic implies that even underlack of commitment of the buyer and unobservable investment, the seller’s equilibriuminvestment is positive. Since we know that the unobservability of the investment is anecessary condition to avoid the hold-up problem, in this section we consider the limitwhere uncertainty vanishes.

Let’s note first that our model accommodates to the case where the investment bythe seller is socially efficient. Indeed, even though Assumption 1 implies that investingis inefficient for the seller, it may be efficient for the buyer. For example, if u is twicedifferentiable, one can easily compute the drift of u(v) by using stochastic analysis,which is given by:

µu(v) = µ(v)u′(v) + 12σ(v)2 u′′(v) .

In this case, if µu(v) > c(v), investing is socially efficient.

Proposition 3.2. Re-parametrize σ(v) by λ σ(v), with λ > 0 and some fixed σ(v).Then, as λ 0, the equilibrium investment vanishes, that is, v−, v+ → v0.

Proposition 3.2 establishes that not only unobservability is important for inducinginvestment, but also randomness. Intuitively, even though investing is on average ineffi-cient from the seller point of view, she invests in order to keep the good if the investmentgoes well, and sell it otherwise. In our model, the increase on the expected value ofinspecting given by option of selling the good compensates the cost of experimentation.Still, if the volatility σ(v) is small, aggressive pricing discourages investment.

3.2 (Bertrand-) Competitive Market

All results presented so far rely on the assumption that there is only one buyer inthe market. In order to understand the effects of private investment by a seller in acompetitive environment we consider the opposite case of a monopsonist: many buyers(Bertrand-)compete in order to buy the good.

Assume that, only for this section, the seller faces at least two homogeneous buyersin the second stage, who simultaneously make offers to the seller. As it is well known,Bertrand-competition in the second stage drives the expected payoff of each buyer to0. Still, since our distribution of seller’s valuations is endogenous, it is unclear whetherthe price offered by the buyers in equilibrium is unique or they will randomize using adistribution with non-degenerated support.

Let Fp now denote the distribution of the highest price offer made in equilibrium.All offers in the support of Fp (denoted P) give a payoff equal to 0 to a buyer. Thefollowing proposition characterizes the equilibria in the competitive case:

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Proposition 3.3. Fix an equilibrium. There exist some v−, v1, v+ ∈ R such that

1. u(v−) < v1, V = v− ∪ [v1, v+] and P = u(v−) ∪ [v1, v+],

2. Proposition 3.1 applies in [v1, v+],

where we use the convention that [a, b] = ∅ if a > b.

In our model, perfect competition may not lead to a single price being offered. Whenv1 ≤ v+ (which happens, for example, when u(v0) is close to v0), our equilibria featurestwo disconnected regions in the support of prices and valuations.

In any equilibrium perfect competition imposes that inf P ≥ u(inf V), so clearlyV 6= P . Then, differently from the monopsonist case discussed before, there is a “lowestprice” (equal to u(v−)) which only attracts the seller when her valuation reaches theminimum of the support of her equilibrium expected valuations.

Clearly, if u(v) = u for all v ∈ R, for some u ∈ R, the only possible competitiveequilibrium trade outcome involves only one price, P = u. This corresponds tov1 > v+ in Proposition 3.3. Therefore, following Remark 3.1, in the only competitiveequilibrium trade outcome in the experimentation model of Section 2, only price −g isoffered, and no experimentation takes place.

4 Conclusions

Introducing the possibility of gathering private information by trading counterpartsmay have an important impact on trade outcomes. If buyers can experiment in orderto learn how a given product fits their needs, they are going to be able make betterpurchasing decisions. Similarly, if sellers’ investment in their products is successful,they are going to be less willing to sell their goods at a low price.

The effect of pre-trade endogenous information far from trivial. In equilibrium,more information may lead to less information rents, since the uninformed party maybecome more aggressive. As a result, even when the cost of acquiring information isarbitrarily cheap, we find that the amount information that the buyer decides to obtainis bounded away from full information. This implies that, depending on the gains fromtrade, there may be more or less trade than in a model with a fully informed buyer.An intermediate cost of experimentation may help the buyer to credibly commit not toacquire too much information, and then obtain some information rents.

Our work highlights the importance that the uncertainty of investment outcomes hason trade. Even if investments are inefficient, the fact that the price offer is independentof the outcome of the investment induces sellers to over-invest. They know that if theinvestment is successful they can retain the good, while if it is unsuccessful they can

18

sell the good to the buyer. As the randomness decreases, investment does so.

A Omitted Proofs

A.1 Proof of Lemmas 2.1 and 2.2 and Proposition 2.1

In order to avoid redundancies in the proofs, we prove Lemmas 2.1 and 2.2 and Proposition2.1 under the general investment process introduced in Section 3. It is easy to adapt themfor the experimentation model (see Remark 3.1). The reader is urged to read Section 3 beforereading these proofs.

Proof of Lemma 2.1

Proof. Sequential rationality trivially imposes that α(p) = 1 if p > vτ and α(p) = 0 if p < vτ .Furthermore, the standard underpricing argument implies that Pr(vτ = p) > 0 for some pthen α(p) = 1. As a result, for each p, α and α only differ at a history where vτ = p if theequilibrium probability of vτ = p is 0, so they generate the same outcome distribution. Sinceα is sequentially rational, the result holds.

Proof of Lemma 2.2

Proof. Assume, by contradiction, that |V| = 1, and let v∗ be such that V = v∗. FromLemma 2.1 it is clear that the best response by the buyer is to offer v∗ if v∗ ≥ u(v∗) or anunacceptable offer, and therefore the gains from trade for the seller are 0. Therefore, by theassumption of that investment is not efficient for the seller (Assumption 1), she is better offnot investing at all. So, v∗ = v0, the buyer offers v0 for sure and the seller does not exert anyeffort.

The seller may deviate and, instead, exert effort until the value vt hits some values v−and v+, with v− < v0 < v+. Note that if the seller stops investing at some time value v shereceives a payoff of maxv, v0. Therefore, for v− and v+ close enough to v0, the value functionfor valuations in v ∈ [v−, v+] is characterized by equation (3.1) with boundary conditionsV (v−) = v0 and V (v+) = v+. Standard stochastic analysis shows us that since the value fromstopping at v0 has a kink, it is never optimal to not invest.

Proof of Proposition 2.1

We divide the proof in five lemmas. In the first, we prove P ⊂ V. In the second, the strictconvexity of V . The third claims that P has no gaps. In the fourth, we show P ⊃ V,so P = V. Finally, we present a fifth lemma which characterizes the mass points of theequilibrium distributions.

Lemma A.1. P ⊂ V.

19

Proof. Note that any price that is accepted with positive probability in equilibrium mustbe equal or lower than v+, since otherwise the buyer can lower his offer while ensuring thesame acceptance probability. Note also that v+ > v0 since otherwise the highest equilibriumpayoff of the seller in the second stage is v+. Since she incurs experimentation costs, notexperimenting is a profitable deviation.

Similarly, v− ≤ v0.9 In order to see this, assume that v− > v0. The payoff in the secondstage if vτ = v is equal

V (v) = Pr(p > v)E[p|p > v]︸ ︷︷ ︸sells the object

+ Pr(p ≤ v) v︸ ︷︷ ︸retains the object

. (A.1)

As a result, we have the following inequality

V (v−)−V (v0) ≤ Pr(p≤v−) (v−−v0) .

Let C > 0 be the expected total cost induced by the stopping time given by inft|vt = v−(if it is +∞ then we have a contradiction and v− ≤ v0). Since by Assumption 1 investing isinefficient for the seller, necessarily v−−C < v0, so v−−v0 < C. As a result, V (v−)−V (v0) < C,which implies not experimenting is a profitable deviation.

Since v− ≤ v0 and u(v) > v for v ≤ v0, the payoff of the buyer is strictly positive in anyequilibrium (offering v− + ε for some small ε > 0, for example, gives him a positive payoff).As a result, inf P ≥ inf V. Assume then that there is some p ∈ P such that p /∈ V. Thisimplies that there exists some ε > 0 such that [p−ε, p]∩V = ∅, and p−ε > v−. Offering pricep is dominated by offering p − ε, since the second one is lower and the probability of beingaccepted is the same. So, P ⊂ V.

Lemma A.2. The continuation value of the seller V is strictly convex in [inf V, supV].

Proof. Denoting v− ≡ inf V and v+ ≡ supV, Lemma A.1 and standard optimal-stoppinganalysis imply that the value function of the seller is twice-differentiable and solves (3.1) in[v−, v+] the with boundary conditions V (v−) = E[p] and V (v+) = v+. Since V (v) = E[p]

for all v < v− and V (v) = v for all v > v+, the standard smooth-pasting arguments implyV ′(v−) = 0 and V ′(v+) = 1.

Using equation (3.1), the boundary conditions and Assumption 1 it is clear that V ′′(v−) >

0 and V ′′(v+) > 0. Assume that V ′′(·) reaches negative values in (v−, v+) and let v′ ≡maxv|V ′′(v) = 0. Note that from equation (3.1) we have that V ′(v) > 1 and V ′′(v) > 0 forall v ∈ (v′, v+). This contradicts the smooth pasting condition at v+.

Lemma A.3. P has no gaps.

Proof. Note first that for any v ∈ V, since the seller is willing to stop investing, her continuationvalue at v is given by (A.1). Hence, we have that, for any v1, v2 ∈ V, with v1 < v2,

V (v2)−V (v1) = Pr(p≤v1) (v2−v1)− Pr(p∈(v1, v2])E[v2−p|p∈(v1, v2]] .

9Note that this result is trivial in the experimentation setting of Section 2 because the drift of vt is 0and because of Lemma 2.2.

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It is then clear that if v ∈ V and v is not isolated, then V ′(v) = Pr(p≤v). In particular, thisimplies that v+ is not an mass point of Fp, since the fact that V is twice-differentiable impliesthat limvv+ V

′(v) = 1.

Assume, by contradiction, that p1, p2 ∈ P, with p1 < p2, but (p1, p2) 6⊂ P. Take anyv ∈ (p1, p2). Since P ⊂ V and vt is continuous, v ∈ [inf V, supV]. Then, we have

V (v) ≥ V (p1) + Pr(p≤p1) (v−p1) ≥ V (p1) +V (p2)−V (p1)

p2 − p1(v−p1) .

This contradicts the strict convexity of V in [inf V, supV].

Lemma A.4. P ⊃ V.

Proof. Assume first, by contradiction, that inf V < inf P. In this case, since inf P ∈ V, wehave both V (inf V) = E[p] and V (inf P) = E[p]. This contradicts that V (v) ≥ E[p] and V isstrictly convex in [inf V, supV].

Assume now, also by contradiction, that supV > supP. In this case, since supP ∈ V, wehave both V (supV) = supV and V (supP) = supP, which are the corresponding autarchyvalues. This is incompatible with Assumption 1, which that the seller does not invest inisolation.

Since P has no gaps and P ⊂ V, the result holds.

Lemma A.5. Fp does not have mass points and Fv’s only mass point is v−.

Proof. Let’s assume first that there is a mass point of Fp at some p ∈ (v−, v+). This impliesthat V ′(·) has a kink at this point, which is incompatible with the fact that V is twice differ-entiable. Note Fp does not have a mass point at v− since, otherwise, limvv− V

′(v) = Pr(v >

v−) > 0. The proof of Lemma A.3 shows that Fp cannot have a mass point at p = v+. Assumethat there Fp has a mass point at p = v−. Using (A.1) we have

V (v) = E[p] + Pr(p ≤ v) (v−E[p|p≤v]) ≥ V (v−) + Pr(p=v−) (v−v−) .

This is a clear contradiction with the boundary condition V ′(v−) = 0.

Assume finally that there is a mass point of Fv at some v′ > v− with mass m. In this case,the payoff of the buyer from offering v′ − ε is Π(v′) −m (v(v′) − v′) + O(ε), if ε > 0 is smallenough. This contradicts the indifference condition of the buyer. Clearly, Fv has a mass pointin v− in order to ensure that the buyer is willing to offer v−.

A.2 Proof of Proposition 2.2

Proof. The form of the distributions Fv and Fp is a direct consequence of Lemma A.5 (recallthat it is written in the setting of Section 3) and the indifference conditions of the seller andthe buyer. So, only is left to show that there exists a unique equilibrium.

The proof of the existence of an optimal strategy that generates Fv follows from Ankirchner,Hobson and Strack (2014). They show (Theorem 2) that if E[v|Fv] = v0 (which is satisfied by

21

construction) then Fv can be optimally embedded (i.e., there exist an cost-minimizing stoppingtime rule) from the degenerated (Dirac’s delta) function δv0 if Fv ∈ L1. Also, the expectedstopping time (and then total cost) of such optimal strategy is finite.

Since (vt)t is a martingale, v0 = E[vτ ]. As a result, for each fixed v− and v+, we can usethe explicit distribution of valuations to compute the implied average valuation of the buyer,which leads us to

v0 =

∫ v+

v−

v− + g

(v + g)2v dv +

v− + g

v+ + gv+

= v− + (v− + g) log

(v+ + g

v− + g

), (A.2)

where v−+g(v+g)2

inside the integral is the implied PDF of Fv for v ∈ [v−, v+). Then, for a fixedv0 and v−, let vFv

+ (v−) solve the equation (A.2) for v+, so it satisfies

vFv+ (v−) ≡ −g + (g + v−) exp

(v0 − v−g + v−

), (A.3)

where the super-index Fv in vFv+ denotes that is found using the value distribution. It is easy

to see that vFv+ (·) is strictly decreasing. In particular, as v− v0, we have v+ v0. Also it

is easy to show (see Proposition 2.5) that either there exists v− ∈ [max−g, vL, v0] such thatvFv

+ (v−) = vH or vFv+ (vL) ∈ (v0, vH).

As it is obvious from the distribution of Fp, for each v− there exists a unique v+ = vFp

+ (v−)

such that Fp(v−) = 0 and Fp(v+) = 1. Also it is clear that if vFp

+ (·) is strictly increasing,limv−vL v

Fp

+ (v−) = vL and limv−vH vFp

+ (v−) = vH .

The previous two paragraphs make evident that a unique solution to vFv+ (v−) = v

Fp

+ (v−)

exists.

A.3 Proof of Proposition 2.3

Proof. We now make use the notation used the end of the proof of Proposition 2.2. Differ-entiating the right hand side of equation (A.3) with respect to g (while keeping v− constant)gives

∂vFv+ (v−)

∂g= (1− x) ex − 1 ,

where x ≡ v0−v−g+v−

. The function in the right hand side of the previous expression is alwaysnegative for x > 0, so the whole curve vFv

+ (·) shifts down when g increases. As a result, thesolution to vFv

+ (v−) = vFp

+ (v−) decreases, and since vFp

+ (·) remains unchanged and is increasing,the equilibrium value of v+ also decreases.

As g →∞, it is easy to show from equation (A.3) that v+ v0. In this case, the mass ofFv at v+ approaches v−+g

v++g , which itself approaches 1.

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A.4 Proof of Proposition 2.4

Proof. Using equation (A.3) it is easy to see that when v− increases then v+ decreases. Sincewhen c the density of prices given in proposition 1 increases point-wise, necessarily v− isincreasing in c, so v+ is decreasing in c. Since v0 ∈ [v−, v+] when c → ∞ both tend to v0.Finally, since −g < v−, it is clear that when both v− and v+ are very close to v0, it is optimalfor the seller set a price that ensures trade with high probability.

A.5 Proof of Proposition 2.5

Proof. Note that c→ 0 implies that, for each given v, the price PDF F ′p(v) in Proposition 2.2tends to 0. Given that Fp does not have mass point, this implies that either v− → vL andv+ → vH or both. Note that, for each c > 0, v− and v+ satisfy equation (A.2). It is easy toverify that if g < g, then necessarily v+ → vH when c→ 0, and the limit for v− solves equation(A.2) with v+ = vH , which is the equation in the statement of the proposition. Alternatively,g > g, then necessarily v− → vL, and the limit for v+ when c→ 0 solves equation (A.2) withv− = vL, which gives us the expression in the statement of the proposition.

Since, generically, either v− → vL or v+ → vH , the price distribution is generically degen-erated towards vL and towards vH , respectively. In the first case, since Fv does not accumulatemass close to vL, the probability of trade necessarily converges to 1. In the second case, inthe limit, the probability of tends to the mass of Fv at v+, which, using the formula for Fv inProposition 2.2, equals g+v−

g+vH.

It is only left to show that m ≡ g+v−g+vH

< φ0. Simple algebra shows that the right handside of equation (A.2) is increasing in v− and in g when v+ = vH . We can replace v− bymvH − (1−m) g in equation (A.2). This defines an implicit function g(m) which is given by

g(m) ≡ −v0 −mvH(1− log(m))

1−m+m log(m)⇒ g′(m) =

−(vH − v0) log(m)

(m−m log(m)− 1)2> 0 .

So, m reaches its maximum when g = g. Since g is found solving equation (A.2) with v− = vL

and v+ = vH , vL+gvH+g is the maximum value that m can reach. Finally, using equation (A.2)

again we can find

φ0vL+gvH+g

=vH + g

vH − vLlog

(vH + g

vL + g

).

Simple calculus shows that the right hand side of the previous equation is greater than 1 forany g > −vH .

A.6 Proof of Proposition 2.6

Proof. Trivial using the arguments in the text.

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A.7 Proof of Proposition 3.1

Proof. This proposition follows immediately form Lemma A.5 and the indifference conditionsof the seller and the buyer (v0 ∈ [v−, v+] is proven in Lemma A.1).

A.8 Proof of Proposition 3.2

Proof. As we know from Lemma A.2, in any equilibrium, V ′(v) ∈ [0, 1] for all v ∈ R. SinceV ′′(v) = 0 when v /∈ [v−, v+], using the HJB equation (3.1) we have that

V ′′(v0) =c(v)− µ(v)V ′(v)

12σ(v)2

≥ 1

λ2

c(v)− µ(v)V ′(v)12 σ(v)2

Since V ′(v) ∈ [0, 1] and c(v) > µ(v), clearly V ′′(v0)→∞ when λ 0. Since V ′(v−) = 0 andV ′(v+) = 1 we have that v− v0 and v+ v−.

A.9 Proof of Proposition 3.3

Proof. Fix an equilibrium. Note that the infimum of the price support is no lower than u(v−),since otherwise buyers would earn a positive payoff. If the seller stops investing at somev ∈ (v−, u(v−)) provides her payoff is E[p]. Given the strict convexity of V between v− andv+, (v−, u(v−)) ∩ V = ∅.

Let’s define v1 ≡ infv ∈ V|v > v−, so by the previous argument v1 ≥ u(v−). IfP∩[v−, v1) = ∅ then V (v1) = E[p] = V (v−), which contradicts the strict convexity of V . Hence,necessarily, u(v−) < v1 and, given the 0-payoff condition for the buyers, [v−, v1)∩P = u(v−).

Let’s now define p1 ≡ infp ∈ P|p > u(v−). Clearly, p1 ≥ v1. If p1 > v1 then necessarilyv1 is an isolated point of V. Indeed, otherwise, using similar arguments as in Lemma A.3,there should be elements of P arbitrarily close to v1, which is a contradiction. Nevertheless,if v1 is an isolated point of V (and therefore a mass point), offering p = v1 gives to a buyer astrictly positive payoff, which is a contradiction. Therefore, p1 = v1. Furthermore, competitionimplies E[u(v)|v ≤ v1] = v1, so v1 is a mass point of Fv.

Since v1 ∈ P ∩ V and any price above v1 can be the highest price offered to the sellerwith positive probability, all should give 0 payoff to the seller. Therefore, we can easiltyadapt the previous analysis (Lemmas 2.1 and 2.2 and Proposition 2.1) and find that P =

u(v−) ∪ [v1, v+], and V = v− ∪ [v1, v+].

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