Dynamic quality signaling with hidden actions

Games and Economic Behavior 113 (2019) 116–136

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Games and Economic Behavior www.elsevier.com/locate/geb

Dynamic quality signaling with hidden actions ✩ Francesc Dilmé University of Bonn, Germany

a r t i c l e

i n f o

Article history: Received 25 September 2015 Available online 26 September 2018 JEL classification: D82 D83 C73 J24 Keywords: Dynamic signaling Dynamic moral hazard Endogenous effort

a b s t r a c t A seller owns a firm and privately knows its underlying value. She can exert a costly hidden effort over time to affect a noisy signal. Arriving competitive buyers use the history of signals to infer the value of the firm and make price offers. We show that the degree of efficiency of sending highly-informative signals plays a crucial role in shaping how information is transmitted. If exerting a high signaling effort is efficient, the seller of a high-value firm responds to bad luck by increasing the level of effort and, with a high probability, she sells the firm at a high price. In contrast, when high signaling effort is inefficient, the seller stops exerting signaling effort when buyers become pessimistic about the value of the firm, and she sells it at a low price or retains it for herself. In both cases, the equilibrium effort level is inefficiently low. © 2018 Elsevier Inc. All rights reserved.

1. Introduction When an entrepreneur wants to sell a firm, she often possesses private information about its underlying value. This information includes some characteristics of the firm, such as the value of its patents or its growth potential in different markets. In addition, some actions undertaken by the entrepreneur may be unobservable to potential buyers, such as the amount of personal funds devoted to keeping the firm running, or the time spent in managerial and other activities. Potential buyers of the firm use, instead, information that becomes available over time (for example, sales data) to infer the unobserved choices and value of the firm, which affects their willingness to pay and the resulting price offers. Our main goal is to understand, in the previous setting, how the idiosyncratic characteristics of the firm interact with the endogenous choices of the entrepreneur, and how these shape the buyers’ interpretation of the noisy information that becomes available to them. We focus on how the endogenous informativeness and efficiency of the signaling process are shaped by the structure of the cost of signaling. The paper analyzes a model in which a seller of a firm has private information about its underlying value (or type), which can be either low or high. The seller of a high-value firm can exert a hidden signaling effort to influence a stochastic process, which takes the form of a diffusion process, interpreted as sales. The cost of exerting the effort has both a fixed and a variable component. Competitive buyers use the history of signals to make inferences about the value of the firm, and make private offers to the seller to purchase it. The game ends when the seller accepts one such offer. Only the trade of the high-value firm is efficient.

✩ Former versions of this paper used the title “Dynamic Quality Signaling with Moral Hazard.” I am grateful to Stephan Lauermann, George Mailath, Andy Postlewaite, Jan-Henrik Steg, Philipp Strack, the associate editor and two referees for their helpful comments. E-mail address: [email protected].

https://doi.org/10.1016/j.geb.2018.07.010 0899-8256/© 2018 Elsevier Inc. All rights reserved.

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We characterize the set of Markov equilibria of our model using the posterior about the value of the firm being high as the state variable, and we focus our analysis on equilibria where the range of posteriors where the signal is informative is maximal. Owing to the lack of gains from trade for the low-value firm, the different types of the seller accept and reject the same equilibrium offers. This leaves effort as the only source of separation between the types of the seller, and hence allows a clean analysis of the effects of hidden actions on dynamic signaling. So, in equilibrium, the seller of a high-value firm exerts a costly effort to make buyers more optimistic about the value of the firm, hoping to receive a high offer in the future. Still, since buyers cannot observe such an effort, the effort that buyers expect in equilibrium must equal the optimal effort choice given these beliefs. We show that the equilibrium signaling effort is inefficiently low due to its unobservability. To illustrate this, we divide the analysis into two parts. First, Sections 2.1 and 3 characterize effort choices that are part of equilibrium behavior; that is, the ones that when buyers believe that the seller chooses them, then the seller has the incentive to choose them. Then, Section 4 computes the seller’s payoff for each strategy, assuming both that she chooses it and the buyers believe that she chooses it. In the second part of the analysis, we show that if the cost function is not very convex, the total signaling waste decreases when the signaling effort increases; but when the cost function is highly convex, an intermediate signaling effort is optimal. In both cases, the equilibrium effort exerted by the seller of a high-value firm obtained in the first part of the analysis is lower than the optimal effort obtained in the second part. The reason is that, given that the effort is unobservable, the seller is unable to internalize the benefit that increasing the effort choice generates. We obtain two kinds of intertemporal equilibrium signaling strategies, depending on whether sending highly informative signals is efficient or not. When the variable cost function is highly convex, exerting a high effort is very costly, and the presence of a fixed cost limits the seller’s willingness to provide lowly informative signals. In effect, we obtain a “give-up effect”: when buyers are pessimistic about the value of the firm, the seller lowers her signaling effort, which makes signaling more inefficient and increases the likelihood that she retains the firm for herself. Intuitively, revealing the (high) value would require providing a large amount of information, which is too costly given the high convexity of the cost function. Alternatively, when the variable cost function is not very convex, exerting a high effort is more efficient than exerting a low effort. Still, given that the effort is unobservable, an equilibrium high effort must be incentive-compatible given the beliefs of the buyers about the effort choice. As a result, the equilibrium effort is found to be high (and more efficient) only when buyers are pessimistic, since then a small increment of the posterior generates a big increase in the equilibrium probability of getting a high offer. In this case we obtain that firms with a low market valuation perform better than firms with an intermediate market valuation: the high equilibrium effort of the seller compensates the composition effect. Our model can be applied to education. Indeed, the education process is by nature dynamic since the information about students’ skills is progressively revealed over time. Grades stochastically depend on the (skill-adjusted) effort of the students; so students, knowing their skills and grade history, decide how much effort they will exert to affect their future grades. On the other side of the market, employers use grades to infer the students’ productivity and make them job offers. In this setting, if the (utility) cost or the effectiveness of obtaining high grades is correlated with innate skills, students with different productivity levels may exert different levels of effort. The organization of the paper is as follows. After this introduction, we review the related literature. Section 2 introduces our model. The analysis of equilibrium behavior is contained in Section 3. In Section 4 we analyze the observable effort case. Section 5 concludes. An appendix contains the proofs of all lemmas and propositions of the previous sections. 1.1. Literature review Our paper contributes to the literature on dynamic signaling with noisy signals, which studies the effect of news about the seller’s type on trade outcomes. Kremer and Skrzypacz (2007) analyze the arrival of news about the firm at some fixed time in a version of Swinkels (1999)’s model. Daley and Green (2012) and Kolb (2015, 2018), instead, consider public stochastic signals related to the type. Papers on adverse selection such as Lauermann and Wolinsky (2016) and Kaya and Kim (2018) consider a lemons environment similar to the private offers case of Hörner and Vieille (2009), allowing for private signals. As in the previous papers, we consider a signal that is payoff-irrelevant for the buyers, but we take a step further by endogenizing its informativeness. In an independent work, Heinsalu (2017, 2018) discusses two signaling models with endogenous effort. In the first, he analyzes equilibria where the effort exerted is independent of the history. In the second, he characterizes “reversed effort” equilibria, that is, equilibria where the high-cost seller exerts higher effort than the low-cost seller. Relative to the mentioned papers, the focus of our paper is on the dynamics of the speed of learning, the effect of the signaling efficiency on trade outcomes and the new trade-offs that the history-dependent informativeness of the signal implies for the seller. We show that the efficiency of the signaling effort crucially affects the distribution of transaction prices and the monotonicity of the endogenous signal informativeness. We also determine that the amount of equilibrium signaling is always inefficiently low due to the unobservability of the effort. Static noisy signaling was introduced by Matthews and Mirman (1983) in a limit pricing model. They show that adding noise to the signal reduces the equilibrium set. de Haan et al. (2011) study a model similar to the analogous “stage game” of our dynamic model. One of our main goals is to analyze how the form of the flow cost (interpreted as the cost in the “stage game”) affects the dynamic incentives of the sender, the dynamics of the (endogenous) informativeness of the signal and the solution of the fixed point between the effort that the buyers believe the seller exerts and the optimal effort of the seller given these beliefs.

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2. The model The model is set in continuous time with infinite horizon, t ∈ R+ . Its main ingredients are the following. Players: The players in the game are one (female) seller and a continuum of short-lived (male) buyers. The seller owns an asset (or firm) and is long lived. The quality (type/value) of the asset may be either low (L-asset/L-seller) or high (H -asset/H -seller). Valuations: The θ -seller values her own asset at U θ , with U H > U L > 0, which is interpreted as the value of retaining the asset for herself (i.e., not selling it). The value of an asset with a given quality is common across all buyers. They value an H -asset at  H ≡  > U H , while their valuation of an L-asset is set at  L = 0.1 Price offers: At each instant, the seller encounters two or more short-lived buyers, who make her secret price offers. We proceed similarly to Faingold and Sannikov (2011) and Daley and Green (2012) and we do not directly model each buyer. Instead, we model the buyers’ offers using an offer function W : [0, 1] → [0, ] indicating, for each posterior of the buyers about the quality of the asset being high, the (highest) offer that buyers make to the seller. As pointed in Daley and Green, this can be microfounded by interpreting W ( p ) as the highest offer that the seller receives from (two or more risk-neutral and expected-payoff maximizer) short-lived buyers when their belief about the asset’s quality being high is p. The equilibrium conditions on W (see Definition 2.1 below) ensure that W reproduces the outcome of the Bertrand competition among short-lived buyers as in, for example, Swinkels (1999). Seller’s strategies: The seller decides which price offers to accept and how much effort to exert in order to signal the quality of her asset. As it is common in settings where the only payoff-relevant variable for the uninformed part of the market is the type of the informed part, we will focus our attention on Markov strategies with the belief about the type of the seller being high as the state variable. Hence, an acceptance strategy for the θ -seller is a closed subset of [0, 1], denoted A θ ⊂ [0, 1], indicating the set of posteriors at which the θ -seller sells her asset. An effort-choice strategy for the H -seller is a 1 function e H : [0, 1] → R+ , indicating the effort that the H -seller exerts at each posterior, such that e − H (0) closed and e H is 1 2 in Lipschitz continuous in [0, 1]\e − H (0). The L-seller, instead, cannot exert effort, so an effort-choice strategy for the L-seller 3 is the function e L ≡ 0. We call a pair (e θ , A θ )θ∈{ L , H } a strategy profile.

Information and beliefs The public belief ( P t )t is a [0, 1]-valued stochastic process adapted to the filtration F B ≡ {FtB }t ≥0 generated by a standard Brownian motion B on a given probability space {, F B , P}. In this section we establish the dynamic equations that ( P t )t satisfies that allow us to interpret it as the posterior of the buyers about the type of the asset being high. We initialize P at some value p 0 ∈ [0, 1], which is interpreted as the common prior that the buyers have at time 0 about the type of the seller being H . For a given effort-choice strategy of the seller eˆ and a belief process ( P t )t , there is a public (noisy) signal X about the effort exerted by the seller. The public signal satisfies

d X t = eˆ ( P t ) dt + σ dB t . To determine the evolution of the belief process, fix a strategy profile (e θ , A θ )θ∈{ L , H } believed by the buyers, and an

ˆ ). Assume first that p 0 ∈ actual strategy by the seller (ˆe , A / A L ∪ A H . In this case, buyers learn from the public signal, since not accepting the equilibrium offers is not informative about the type. So, for all t < τˆ such that P t ∈ / A L ∪ A H , we assume that P satisfies the following stochastic differential equation, often called filtering equation:

˜ (ˆe ( P t ), P t , e H ( P t )) dt + σ˜ ( P t , e H ( P t )) dB t dP t = μ

(1)

where for all p, eˆ ( p ) and e H ( p ) we have

μ˜ (ˆe( p ), p , e H ( p )) ≡ σ˜ ( p , e H ( p )) ≡

(1 − p ) p e H ( p ) (ˆe ( p ) − p e H ( p ))

(1 − p ) p e H ( p )

σ

σ2 .

and

(2) (3)

1 Section 2.2 shows that θ , for θ ∈ { L , H }, can be rationalized as coming from a type-dependent flow payoff form holding the θ -asset in the limit where discounting goes to 0. Note that we assume that there are gains from trade only from the H -asset. As we will see, the main implication of this assumption is that effort is the only equilibrium source of transmission of information about the type, and not the rejection of offers (see Lemma 2.1 and footnote 9). 2 The conditions on e H simplify the technical analysis of our model. Still, the set of functions satisfying them is rich enough that any piece-wise continuous function from [0, 1] to R+ (or function approachable with piece-wise continuous functions) can be approximated using effort-choice strategies of the H -seller. 3 The assumption that one type is “handicapped” is common in the reputations literature (see, for example, Mailath and Samuelson (2001) or Hörner (2002)). In our model, the handicapped type acts strategically through accepting or rejecting offers, as in the standard models of dynamic signaling.

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Appendix B derives these equations, and establishes the uniqueness of a strong solution. Consider now the case where p 0 ∈ A L ∪ A H . If p 0 ∈ A L \ A H then the rejection of an offer reveals the type of the seller to be H , so in this case P 0+ = 1. If, on the contrary, p 0 ∈ A H \ A L , then rejection convinces the buyers that the type of the seller is L, so in this case P 0+ = 0. Finally, if p 0 ∈ A L ∪ A H and the seller (off the path of play) rejects the offer, we can not use the Bayes’ rule to update the beliefs. In this case we assume, for convenience and without loss of generality, that the posterior after the rejection remains p 0 .4 So, the posterior at (t , ω) ∈ R+ ×  (conditional on t ≤ τˆ (ω)) is given by

P t (ω) =

⎧ ⎪ ⎨0

if t > τ L (ω)> τ H (ω) ,

P τ H (ω)∧t (ω) if τ L (ω)= τ H (ω)< t or τ L (ω)∧ τ H (ω)≥ t , ⎪ ⎩ 1 if t > τ H (ω)> τ L (ω) ,

(4)

where P τ H (ω)∧t (ω) is equal to the solution of (1) at time min{τ H (ω), t } when the outcome is

ω ∈ .

Payoffs Exerting effort is costly for the seller. For notational convenience, we assume that the L-seller has a flow cost from exerting effort e ∈ R+ equal to c L (e ) = Ie>0 ∞, so choosing effort equal to 0 (at 0 cost) is always optimal. The H -seller, instead, exerting effort e incurs a flow cost c H (e ) given by

c H (e ) ≡ Ie>0 c 0 + C H e α ,

(5)

with c 0 , C H > 0 and α > 1.5 Fix an offer function W and a strategy profile (e θ , A θ )θ∈{ L , H } inducing stopping times

τL and τ H . The payoff for the

θ -seller of following a strategy (ˆe , Aˆ ) inducing a stopping time τˆ is composed of the flow cost of providing effort and the lump-sum payoff when the game stops6

 ˆ   τˆ Eeˆ , A − t c θ (ˆe ( P s )) ds + Iτˆ <∞ W ( P τˆ ) + Iτˆ =∞ U θ .

(6)

It will be useful to denote the value of following the equilibrium strategy as follows



V θ ( p 0 ) ≡ Eeθ , A θ −

 τθ t

c θ (e θ ( P s )) ds + Iτθ <∞ W ( P τθ ) + Iτθ =∞ U θ

(7)

where, with some abuse of notation, we make the dependence on the prior explicit to compute. 2.1. Equilibrium characterization Equilibrium concept Our definition of equilibrium is analogous to that in Daley and Green (2012), and corresponds to a Markov perfect equilibrium. It requires (1) the strategy of the seller to be optimal (given the price process), (2) the transaction price to be equal to the expected valuation of the transacted asset when trade happens (that is, buyers make zero profits due to Bertrand competition), and (3) the continuous-time analogous of no-profitable deviation by the buyers: Definition 2.1. An equilibrium is a strategy profile (e θ , A θ )θ∈{ L , H } and price offer function W such that for any p 0 ∈ [0, 1]: 1. Sellers Optimality. For all θ ∈ { L , H }, (e θ , A θ ) maximizes (6) given W and when the strategy believed by the buyers is (e θ  , A θ  )θ  ∈{ L , H } .7

4 Notice that if the posterior does not change after an unexpected rejection at some p ∈ A L ∪ A H , which can be sustained if e H ( p ) = 0, the seller cannot gain from the rejection itself. As a result, even if this restriction is relaxed (since Bayes’ rule does not apply after an unexpected rejection), if a seller is not willing to deviate for a given strategy profile, she is also not willing to deviate for a strategy profile where p does not change after the deviation. 5 The cost c θ (e ) shall be interpreted as the “net cost” of increasing sales, that is, the difference between the cost and the expected revenue that exerting effort involves. This normalization allows us to treat X as a payoff-irrelevant signal. We can interpret the fixed cost of providing effort to increase sales as an opportunity cost of the time devoted to this. In the education setting, this may be regarded as the cost of attending class (opportunity cost in salaries, for example). Low-type students, instead, could already be enjoying their outside option, by just taking the exams. 6 According to equation (6), the seller does not discount future payoffs. Still, the signal is informative only if the H -seller exerts a positive effort and therefore incurs at least a flow cost of c 0 > 0 so, even though waiting is technically costless, it is also useless. As a result, the tradeoffs that the H -seller faces are similar when time discounting is introduced. Section 2.2 establishes that our results are robust to introducing time discounting by the seller. 7 ˆ θ ) as defined above. Hence, if the seller deviates, the buyers’ Note that we require (e θ , A θ ) to be optimal among all (deviating) Markov strategies (ˆe θ , A belief is computed under the assumption of no deviation, see equation (1).

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2. Buyers’ Zero-Profit. Whenever p 0 ∈ A L ∪ A H we have

W ( p0 ) =

⎧ ⎪ ⎨0 ⎪ ⎩

if p 0 ∈ A L \ A H ,

p0 

if p 0 ∈ A L ∩ A H ,



if p 0 ∈ A H \ A L .

(8)

3. No (Unrealized) Deals.8 V H ( p 0 ) ≥ p 0 . Note that, in any equilibrium, V H ( p 0 ) ≥ V L ( p 0 ) for all p 0 ∈ [0, 1]. Indeed, the H -seller has the option of mimicking the strategy of the L-seller. In this case, the H -seller would obtain the same expected revenue from selling the asset and the same expected signaling cost as the L-seller, and a higher value of retaining the asset for herself. Also, individual rationality implies that V θ ( p 0 ) ≥ U θ for all p 0 ∈ [0, 1] and θ ∈ { L , H }. For a fixed equilibrium, we define the following region in the posterior space: 1 R ≡ [0, 1]\(e − H (0)∪ A L ∪ A H ) .

When p 0 ∈ R the H -seller exerts a positive effort and neither type of the seller accepts the price offer W ( p 0 ). As a result, since R is open, the signal is informative in a neighborhood of p 0 . This motivates calling R the signaling region (of the equilibrium). Non-signaling region We begin our equilibrium analysis by characterizing the continuation play outside the signaling region of an equilibrium. The following lemma establishes an important property of equilibria regarding their signaling region: Lemma 2.1. Let R be the signaling region of an equilibrium, and assume p 0 ∈ / R. Then, 1. if p 0  > U H , W ( p 0 ) = p 0  and the asset is transacted immediately (i.e., p 0 ∈ A L ∩ A H ); / A L ∪ A H ); and 2. if p 0  < U H , e H ( p 0 ) = 0 and the seller keeps the asset forever (so p 0 ∈ 3. if p 0  = U H then either 1. or 2. takes place. As a result, in any equilibrium, the two types of the seller use the same acceptance strategy, so A H = A L . The fact that both types of the seller use the same acceptance strategy is implied by both the assumption that there are no gains from trading the L-asset and the assumption that the seller plays a pure strategy. Indeed, intuitively, the L-seller accepts an offer W ( p ) when the posterior is p only if it is higher or equal to U L > 0. If such an offer is rejected for sure by the H -seller, then a buyer offering W ( p ) obtains 0 − W ( p ) < 0, which leads to a contradiction.9 Note that Pr( P τθ ∈ R ) = 0 for all θ ∈{ L , H }. So, in equilibrium, the game never ends at time t if P t ∈ R. This implies the belief updating in R is driven only by the signal realization, not by the rejection of offers. If, instead, P t ∈ / R, beliefs are not updated on the path of play, either because both types of the seller exert effort equal to 0 and use the same acceptance strategy, or because both types accept the price offer immediately. It is not difficult to see that, in our model, an equilibrium with R = ∅ characterized by Lemma 2.1 always exists. Indeed, for example, it is easy to verify that e H ( p ) = 0 for all p, A L = A L = [U H /, 1] and



W ( p) =

0

if p 0  < U H ,

p

if p 0  ≥ U H ,

is an equilibrium. Section 3 discusses the conditions for the existence of equilibria with a non-empty signaling region. Signaling region Let’s now assume p 0 ∈ R. Since R is open, it is the union of disjoint open intervals. We use ( p , p ) to denote the biggest open interval in R that contains p 0 . More formally,

8 The No (Unrealized) Deals condition is the continuous-time analog of the requirement that there is no profitable deviation by a buyer. Intuitively, if ˆ ∈ ( V H ( p ), p ). Since w ( p ) ≤ V θ ( p ) for all θ ∈ { L , H } (the seller has the option of accepting V H ( p ) < p  for some p, a buyer could made an offer in w ˆ > 0, the offer and obtaining a payoff of w ( p )), such an offer would be accepted by all types of the seller. This would provide the buyer a payoff of p  − w which would give the buyer a strictly positive payoff. 9 The fact that the different types of the seller use the same acceptance decision in equilibrium allows us to focus on the effort as the unique source of separation. Instead, other models without an effort choice, such as in Hörner and Vieille (2009) and Daley and Green (2012), focus their attention on differences on the type-dependent acceptance strategy as the source of separation between types.

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Fig. 1. Example of beliefs paths for a signaling region R = ( p , p ). In the simulation, for the same realization of the Brownian motion, the L-seller (gray line) ends up accepting a low offer or keeping the asset for herself, while the H -seller (black line) ends up accepting a high offer.





p ≡ sup {0} ∪ p ≤ p 0  p ∈ /R



 

p ≡ inf {1} ∪ p ≥ p 0  p ∈ /R .

and

(9) (10)

We know by Lemma 2.1 that the value function of the H -seller at p is w ≡ max{ p , U H } and at p it is w ≡ p  > U H .10 Note that since R is open, p < p 0 < p. Then, if the initial prior lies in the region ( p , p ), since P t moves continuously inside R, the belief process stops only when P t reaches either p or p (where the seller will accept the corresponding price offer or stops exerting effort and keeps the asset for herself). Fig. 1 exemplifies two equilibrium beliefs paths, one for each type of the seller, with the same realization of the Brownian motion. Since the effort of the H -seller is higher than the effort of the L-seller, the value of the realized belief process is also higher. As is usual in the literature on dynamic games in continuous-time, we restrict ourselves to equilibria where V H ∈ C 2 ( R ) ∪ C 0 ( R¯ ). In this case, for a given equilibrium strategy e H (·), the optimal effort choice eˆ H (·) of the H -seller solves the Hamilton–Jacobi–Bellman (HJB) equation, which is given by



˜ (ˆe H ( p ), p , e H ( p )) V H ( p ) + 0 = max − c H (ˆe H ( p )) + μ eˆ H ( p )

1 2



σ˜ ( p , e H ( p ))2 V H ( p ) ,

(11)

˜ and σ˜ are given in (2) and (3), respectively. Note that when the believed effort of the H -seller is 0, e H ( p ) = 0, where μ both the drift and the volatility of the beliefs process are 0, independently of the actual effort choice eˆ H ( p ). This is an important feature of our model that differs from the standard dynamic signaling models: if buyers believe that the signal is uninformative, the seller cannot change the buyers’ beliefs through the signal. The maximization problem (11) is strictly concave in eˆ H ( p ) for eˆ H ( p ) > 0. So, under the assumption that R is a signaling region of an equilibrium,11 we can differentiate (11) with respect to eˆ H ( p ), for eˆ H ( p ) > 0 to get the following first-order condition (FOC): C H α eˆ H ( p )α −1 =

 ( p) e H ( p ) (1 − p ) p V H

σ2

.

(12)

We see that the marginal gain from increasing the effort (right-hand side) is increasing in the effort that buyers predict that the seller does, e H ( p ), since when buyers believe that the effort is high they interpret high signals more favorably. Owing to the convexity of the variable cost function, the marginal cost of effort (left-hand side) is increasing in the actual level of effort, eˆ H ( p ). In equilibrium, the strategy believed by the buyers is optimal for the seller, so eˆ H ( p ) = e H ( p ). Using the FOC (12) we can find an integral expression for V H (·):

p V H ( p) = V H ( p) + p

C H α σ 2 e H (q)α −2

(1 − q) q

dq .

(13)

The proof of Lemma 3.1 below shows that the HJB equation (11) and the FOC (12) imply the following functional form for the equilibrium effort of the H -seller:

10 Note that if it was the case that w = U H , then also w = U H , the payoff of the seller net of costs would be just U H . Therefore, exerting effort would be dominated by not exerting effort at all, which would contradict p 0 ∈ R. 11 Lemma 2.1 (which does not rely on the particular form of c H ) ensures that e H ( p ) > 0 when p ∈ R. Furthermore, if R is the signaling region of an equilibrium, the solution V H (·) of the problem (11) satisfies V H ( p ) ≥ p  as is required in our equilibrium definition.

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 e H ( p) =

K 1 (1 − p )

(2 − α ) p

−

2 c0 C H (2 − α )

1/α (14)

,

where K 1 is a constant to be determined by the boundary conditions on V H (·). Remark 2.1. If there is a solution (e H , V H ) to the HJB equation (11) for a given signaling region (satisfying the corresponding boundary conditions), such a solution is unique. Still, for a fixed signaling region, the equilibrium is not unique for two reasons. The first is that, as is usual in trade models, unacceptable offers are not uniquely determined (the only condition is that W ( p ) ≤ V L ( p ) when p ∈ R). Second, in the non-generic case p 0 = UH , when if UH ∈ / R there is multiplicity on the

price processes consistent with equilibria: W ( UH ) = U H (in this case the asset is transacted for sure) or W ( UH ) ≤ U L (in this case the seller keeps the asset for sure). Nevertheless, generically, there is a unique equilibrium trade outcome, that is, there is a unique equilibrium news/beliefs process and a unique distribution of transaction prices. 2.2. Discounting In this section we show that our results are robust to the introduction of some discounting by the seller. We do this by introducing a discount rate ρ > 0 in the seller’s payoff in (6). We then show that all equilibria in our base model can be approximated by equilibria of the discounting model with ρ small enough. In order for U θ to still represent the value for the seller of type θ ∈ { L , H } of retaining the asset for herself (i.e., not selling it), we assume that the θ -seller receives a flow revenue equal to ρ U θ while she owns the asset. Proposition 2.1. There is an equilibrium of our base model with signaling region R = ( p , p ), with 0 < p < p < 1 if and only if for ρ

any ε > 0 there exists a ρ¯ > 0 such that if ρ ∈ (0, ρ¯ ), there exists an equilibrium with signaling region R ρ and effort e H such that ρ max{d( R , R ρ ), sup p ∈ R ∩ R ρ |e H ( p ) − e H ( p )|} < ε , where d is the Hausdorff distance between sets. 3. Most separating equilibria Dynamic signaling models, in general, feature a high equilibrium multiplicity. In order to focus the analysis on a relevant equilibrium subset, different models use different refinements or selection criteria, typically aimed at selecting equilibria with the most separation between the different types of the seller. In dynamic games with asymmetric information, an important source of equilibrium multiplicity arises from the so-called “belief threats”. They refer to the fact that many strategies can be sustained in equilibrium with the “threat” that, if a deviation by the seller is observed (typically not accepting a given offer), the continuation play is “bad” for her, typically implemented by specifying that buyers perceive such a deviation as being taken by a low type. In our model there is an additional “belief threat” due to the presence of hidden actions: buyers may believe that, after a given history (or for a given posterior belief), the effort of the seller is 0 thereafter independently of her type, and therefore the signal is uninformative. This prevents refinements such as Divinity (Banks and Sobel, 1987), D1 (Cho and Kreps, 1987), Never-a-Weak-Best-Response (used in Noldeke and van Damme, 1990) or Belief Monotonicity (used in Swinkels, 1999, and Daley and Green, 2012) from having a bite in our model. So, we find it convenient to directly focus on equilibria with most separation, that is, equilibria where the signaling region is maximized. We will see that these equilibria have properties similar to those of equilibria that pass the selection criteria used in previous models. In order to guarantee the existence of equilibria with the highest amount of signaling, this section focuses its attention on equilibria with a signaling region contained in ( p − , 1), for some p − ∈ (0, UH ). Such a lower bound on the posterior can be viewed as a threshold where the buyers stop paying attention to the performance of the firm, so (by Lemma 2.1) the seller keeps it for herself. Alternatively, it can also be interpreted as a constraint that lenders impose to the entrepreneur to keep providing her funds to run the firm, so when p ≤ p − the entrepreneur stops running it (and obtains a liquidation value U θ ). We are going to pay particular attention to the behavior of the model when p − is small.12 Definition 3.1. An equilibrium with signaling region R ⊂ ( p − , 1) is a most separating equilibrium (MSE) if R ⊃ R  for any signaling region R  ⊂ ( p − , 1) of another equilibrium. Note that if an MSE exists, all MSEs share the same signaling region, value function and effort choice in R. Then, all MSEs have the same signal distribution and the same distribution over accepted offers. Lemma 3.1. Let V H (·) and V˜ H (·) be the payoff functions of an MSE and a non-MSE, respectively. Then V H ( p ) ≥ V˜ H ( p ) for all p, and V H ( p ) > V˜ H ( p ) for some p.

12

As we will see, the limit as p − → 0 of both strategies and continuation values is well defined.

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Fig. 2. In (a), limit of V H (·) when p − → 0 for an(y) MSE for different values of U H , when α < 2. In (b), the probability of reaching p, for different values of p and p, defined in footnote 15 (gray and black lines correspond to the L-seller and the H -seller, respectively).

The previous result hints that MSEs are in the spirit of most selection criteria, which prevent punishing deviations that are “more likely” to be carried out by high types.13 Indeed, Lemma 3.1 establishes that an(y) MSE is the “most preferred” by the H -seller, in the sense that, for each posterior, she obtains the highest continuation value among all equilibria. Therefore, an MSE solves the H -seller’s problem (11) allowing the seller to choose the boundaries of R (and requiring V H ( p ) = max{U H , p } for p ∈ / R). This naturally leads to the following useful technical result: Lemma 3.2. (Smooth pasting condition) Assume V H is the value function for the H -seller of an MSE. Then, if p − is small enough, V H ∈ C 0 ([0, 1], R) ∩ C 1 (( p − , 1), R). 3.1. Not-very-convex cost function (α < 2) We first study the case where the cost function is not very convex, that is, when establishes the existence of an MSE:

α < 2. The following proposition

Proposition 3.1. Assume α < 2. Then an MSE exists. A signaling region of an MSE is of the form ( p − , p ∗ ) for some p ∗ ∈( UH , 1). Proposition 3.1 states that when α < 2, the signaling region of any MSE always contains ( p − , UH ). This implies that, even if after some “bad luck” the market becomes very pessimistic about the quality of the asset, the H -seller prefers to keep exerting effort in order to avoid having to keep the asset for herself. This is true independently of how small is p − : Fig. 2(a) plots the limit value function for the H -seller in any MSE when p − → 0, for different values of U H , where we see that the H -seller prefers continuing exerting effort than keeping the asset for herself. Our result contrasts with the usual finding in models with a fixed effort (or signal informativeness) like Daley and Green (2012), Kolb (2015, 2018), or Heinsalu (2017), where slow belief updating or high signaling cost (large c 0 and C H ) induce (one type of) the seller to “give up” by accepting an offer. In a dynamic setting with endogenous effort choice, instead, high believed effort increases the speed of belief updating. If the posterior is low, a high believed effort by the buyers is required to make exerting high effort incentive compatible. As we will see in Section 4, the increased efficiency of the signal that exerting high effort implies compensates the high flow cost incurred by the seller. Since when α < 2 the marginal cost of effort increases slower than the marginal gain, such an increase in the effort makes signaling more efficient. Comparative statics We begin with a result about the effect of increasing the noisiness of the signal. An increase of σ lowers the incentive of the seller to exert signaling effort for a given effort level believed by the buyers. Still, in equilibrium, the H -seller has the incentive to exert signaling effort only if the believed signaling effort is higher which, when α < 2, makes it more efficient. As a result, when the cost function is not very convex, increasing the noisiness of the signal increases the amount of information released in equilibrium: Proposition 3.2. Assume α < 2. Let ( p − , p ∗ (σ )) be the signaling region of an MSE for each volatility σ . Then, if σ1 < σ2 , p ∗ (σ1 ) < p ∗ (σ2 ). 13 More formally, assume that if the seller does not trade at a history where both types of the seller were supposed to trade for sure, a posterior is assigned to the deviator and a new equilibrium is played. Then, only MSEs ensure that the L-seller benefits from deviating only if the H -seller also benefits from deviating.

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The intuition for Proposition 3.2 is the following. Since effort is unobservable, the effort that buyers predict in equilibrium has to coincide with best response to such a prediction, which leads to the first order condition (12). When the believed effort is high, buyers interpret high signals as strong evidence of the type being high, so the best response to such a belief is a high signaling effort. Section 4 shows that when α < 2, in the absence of incentive-compatibility constraints, the higher the effort is, the more efficient it becomes: the marginal gain from increasing the effort increases more than the marginal cost when both the believed and actual effort choice increase in the same amount. As a result, an increase in σ is compensated by an increase in the equilibrium effort, which increases the marginal gain from effort more than the marginal cost. In effect, even though there is more signaling waste per unit of time, the H -seller is able to signal the quality of her asset faster (and more efficiently), so the signaling region of any MSE increases.14 Let us finally analyze the properties of any MSE when p − is small. In order to make the following result clear, we write explicitly the dependence p ∗ ( p − ): Proposition 3.3. Assume α < 2 and define p ∗ (0) ≡ lim p − 0 p ∗ ( p − ). Then p ∗ (0) < 1. Also, for all p ∈(0, p ∗ (0)), lim p − 0 e H ( p ) <∞ and lim p − 0 E[τ L | P τL = p − , p 0 = p ] <∞. Proposition 3.3 establishes that, as p − decreases, the upper bound of the signaling region is bounded away from 1, as we can see in Fig. 2(a). To get further intuition, it is useful for our analysis to decompose the value function into two n parts, V H ( p ) = V H ( p ) − E[c H | p ]. The second part, E[c H | p ], is the expected cost of signaling. The first part, V Hn (·), reflects the expected (net) payoff from selling or keeping the asset. Using Bayes’ rule, we find that the expected revenue takes the following form15 n VH ( p) ≡

p ( p − p) p ( p − p)

V H ( p) +

p ( p − p) p ( p − p)

V H ( p) .

(15)

n So, equation (15) shows that an increase in the posterior generates a high increase in the (net) revenue V H ( p ) only when p is small. As a result, exerting a high effort is not incentive compatible when p is not small, and for each given p > 0 the equilibrium effort converges to a finite value (which is the second result of the proposition). Finally, the proposition states that, for a fixed p, conditional on the type of the seller being L, the time it takes for the belief to reach p − (whenever it is reached) remains bounded when we let p − become small. Intuitively, if p − is small and decreases further to some p − ∈ (0, p − ), the effort of the H -seller in ( p − , p − ] is very high (proportional to p −−α , see equation (14)). So, conditional on this region being reached, low realizations of X t lead to fast belief update towards the type of the seller being L.

3.2. Very-convex cost function (α > 2) The following proposition establishes the existence of MSEs when

α > 2:

Proposition 3.4. Assume α > 2. Then an MSE exists. Also, there exist c˜ 0 , c¯ 0 ∈ R+ independent of p − , with 0 < c˜ 0 ≤ c¯ 0 , such that, if p − is small enough, if c 0 ∈ (˜c 0 , c¯ 0 ) the signaling region of an MSE is the union of two disjoint open non-empty intervals, and it is formed by one open non-empty interval otherwise. In Fig. 3(a) we see that when both p − and c 0 are low, the signaling region is close to (0, 1); that is, when p − is small the type of the seller is learned with a high precision in equilibrium. In the limit c 0 → 0, the signaling waste disappears and the H -seller obtains a high offer almost surely, while the L-seller almost surely retains the asset for herself. As c 0 gets larger, R shrinks, since extreme posteriors make belief updating slow enough that the increase in the continuation value does not compensate the cost of exerting signaling effort. When α > 2 high effort is inefficient (see Section 4), so when the posterior is extreme there is not much gain from exerting signaling effort. Using Lemma 2.1, we obtain that either the seller sells the asset (when the posterior is high) or she gives up and retains the asset for herself (when the posterior is low). When, as in Fig. 3(a), UH < p † (where p † is a given threshold defined in the proof of Proposition 3.4 in the Appendix), the rejection region splits into two intervals when c 0 increases above c˜ 0 . One of the intervals contains p † , and the other contains UH . In the first interval fast belief updating makes the signal valuable. This region shrinks as c 0 increases and vanishes when c 0 ≥ c¯ 0 . In the second interval (the lower region) exerting signaling effort remains valuable because, owing

14 de Haan et al. (2011) show that, in a static signaling model with linear costs, high types may increase their effort when the noise increases. In their model, an increase in the noise implies a more wasteful equilibrium signal, and when the level of noise is high enough, only pooling equilibria exist. In our model, when α < 2, the effort increases when the noise increases only when highly informative signals are more efficient than signals with low information content. In this case, the H -seller separates herself from the L-seller more efficiently, so non-pooling equilibria always exist. 15 To see this, note that the proof of Proposition 3.3 proves that the posterior eventually exits the signaling region. Letting πθ ≡ Pr( P τθ = p | P t = p , θ) denote

the probability that P t reaches p if the type of the seller is θ ∈ { L , H }, beliefs consistency requires

πH =

p ( p− p) p ( p− p)

and

πL =

(1− p ) ( p − p ) (1− p ) ( p − p ) . Fig. 2(b) plots

π H and πL .

p 1− p

=

pπ

p

H (1− p ) π L and 1− p

=

p (1−π H ) (1− p ) (1−π L ) , which leads to

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125

Fig. 3. In (a), R (dark gray) as a function of c 0 for a fixed U H , in the limit p − → 0. Immediate trade happens in the white area, while the seller keeps the asset in the light gray area (see Lemma 2.1). In (b), R as a function of U H , for different values of c 0 , also in the limit p − → 0. (The proof of Proposition 3.4 shows that c¯ 0 is independent of U H .)

to the kink in the boundary conditions at UH , the potential gain from signaling is high. As c 0 gets large, this region shrinks but never disappears. In this limit, for most initial beliefs, p 0 , either the asset is sold at p 0  (if p 0 > UH ) or the seller

does exert signaling effort and does not accept any offer (if p 0 < UH ). When, instead, UH ≥ p † , c˜ 0 = c¯ 0 , that is, the rejection region is composed of a single interval for all c 0 . In Fig. 3(b) we plot R as a function of U H , for two different values of c 0 . Again, we see that the higher the cost, the smaller the region where signaling takes place, for any value of U H ∈ (0, ). When c 0 is large (larger than c¯ 0 , represented by the black area), R is an interval containing UH for all values of U H . This interval is small when U H is either low or high (since belief updating is slow), and big for intermediate values of U H , where fast belief updating makes signaling worthwhile. When, instead, c 0 is lower than c¯ 0 (gray area), Proposition 3.4 establishes that, if U H is small enough, R splits into two parts, as explained in the previous paragraph. 3.3. Monotonicity of the effort choice We devote this section to studying the monotonicity of the effort choice of the H -seller. As we will see, the features of the effort choice are qualitatively different in the two cases defined above. We will finally shed some light on which economic forces cause such a difference, and the economic implications it bears. Proposition 3.5. Fix an MSE and let e H and R be its effort choice and signaling region. Then, if α < 2, e H is decreasing inside R, while if α > 2, e H is increasing inside R. In order to obtain some intuition for the different monotonicities of the effort choice established Proposition 3.5, we p define z = log( 1− p ) as the log-likelihood that buyers assign to the type of the seller being H . We can write the first order condition (12) in terms of the log-likelihood as

C H α eˆ H ( z)α −1 =

 ( z) e H ( z) V H

(16)

σ2 z

z

e where, with some abuse of notation, we use e H ( z) and V H ( z) to denote e H ( 1+ ) and V H ( 1+e e z ), respectively. ez  ( z) is necessarily close to 0 when Since V H is increasing and takes values in [U H , ], when p − is small we have that V H  ( z ), z is small, that is, the marginal payoff gain per unit of log-likelihood is small too. When α < 2, for a fixed value of V H the marginal cost of increasing the effort is less than proportional to the exerted effort, but the benefit is proportional  ( z) is close to 0 when z is small, the seller has the incentive to keep exerting effort only to the believed effort. Since V H if belief updating is fast, that is, if the believed effort is high. As we argue in Section 4, a high effort is efficient when α < 2, and therefore the seller is willing to exert it. As z increases, the derivative of the continuation value with respect to z increases too, and therefore the incentive-compatible effort decreases accordingly. When, instead, α > 2, the marginal cost of increasing the effort is more than proportional to e H ( z). This implies that, when the benefit from increasing the log-likelihood is small, the corresponding equilibrium effort is also small. Given that there is a fixed cost of effort, the seller has the incentive to stop exerting effort when p (and z) is small, and as a result the infimum of the signaling region p is bounded away from 0. Proposition 3.5 establishes that some of the main features of the equilibrium belief dynamics and the trade outcome of our model depend qualitatively on the cost structure. These features are most different when the posterior is low, that is, after an “unlucky” realization of the exogenous component of the signal. If, the cost function is very convex, the seller of a high-quality asset responds to bad luck by exerting less effort, which results in she either keeping the asset for herself (so

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Fig. 4. Equilibrium effort at p = 0.5 for R = (0.3, 0.7) as a function of

α.

the pool of unsold assets is not only composed by low quality assets) or transacting it at a low price. When, instead, the cost function is not very convex, the seller of a high-quality asset is aggressive after bad luck, and exerts a high effort in order ensure trading the asset at a high price. In this case, the transaction price is never lower than p ∗ (the price may be higher if p 0 > p ∗ ), independently of the prior. Hence, our model predicts that, the more efficient sending highly informative signals is, the less dispersed prices will tend to be, and the pool of unsold assets will have lower average quality. We can finally analyze how the performance of a firm (in terms of its sales) depends on the convexity of the signaling cost. To do this, we define the unconditional performance of a firm for a given posterior p as the expected drift (given the public posterior) of X t , that is, p e H ( p ). It measures, the expected sales of a firm as a function of its “reputation” (interpreted as the belief about the type of the seller being high). When α > 2, the unconditional performance is increasing in the reputation: a low (high) p involves both a low (high) probability that the type of the seller is H and a low (high) effort from the H -seller. In this case, firms with a high reputation perform better that firms with a low reputation. When, instead, α < 2, the reverse is true when p if is small enough. In this case, even though a low reputation of the firm implies a lower likelihood that the type of the seller is H , it also implies that the H -seller exerts a high signaling effort (proportional to p −1/α ), which compensates the composition effect.16 As a result, if the convexity of the cost of signaling is low, “bad luck reversals” are going to be frequent: after some periods of low sales, firms increase their (expected) sales and, with some probability, recover their reputation. 3.4. The α = 2 case We devote this section to analyzing the knife-edge case α = 2. Even though the study of this case has little economic value per se, it useful in order to understand the differences in the previous results depending on whether α is above or below 2. Notice first that, for any α > 1, the recursive maximization problem (11) is convex in R in the effort choice eˆ H ( p ). So for each believed effort e H (·) and continuation value function V H (·), there is a unique optimal effort choice for the H -seller. For a fixed equilibrium effort e H (·), the marginal gain of increasing the effort (the term in the right-hand side of FOC (12)) is linear in the believed effort e H (·). When the cost function is not highly convex (α < 2), the marginal cost is less convex than a linear function in the actual effort eˆ H (·), which (as we saw in Section 3.1) makes exerting (efficient) high effort incentive compatible when the market is pessimistic. In contrast, when the cost is highly convex (α > 2), high effort is not efficient, so the seller stops exerting effort when the posterior is low to avoid paying the fixed cost (see Section 3.2). In both cases, for a fixed V H (·), there is a unique solution to the equilibrium condition eˆ H ( p ) = e H ( p ) > 0. Assume that c 0 = 0. In this case, when α = 2, the cost, the drift and the squared volatility of the beliefs process are quadratic in (ˆe H ( p ), e H ( p )). This implies that rescaling the effort is equivalent to rescaling time: if both eˆ H ( p ) and e H ( p ) are multiplied by λ > 0, the flow cost is multiplied by a factor λ2 , but also speed of learning is multiplied in this factor. In other words, if eˆ H is a best response to e H then, for any strictly positive parameter λ > 0, λ eˆ H is a best response to λ e H (see the FOC (12)). As a result, when the fixed cost is positive, c 0 > 0, if an equilibrium with positive effort e H existed, the seller would have the incentive to deviate and exert an effort eˆ H > e H . Indeed, exerting an effort e H would a best response to a believed effort equal to e H when c 0 = 0 (since any effort function e H strictly positive in R is an equilibrium effort when c 0 = 0), but the incentive to increase the speed of learning is stronger when c 0 > 0 due to the fixed cost. In consequence, R = ∅ in all equilibria when α = 2 and, as we see in Fig. 4, when α is close to 2, the equilibrium effort exerted by the H -seller is very high. Still, some equilibrium objects of our model behave continuously in the neighborhood of α = 2 (excluding α = 2). To see this, assume that α is close to 2. Fix some p and p, with 0 < p < p < 1 such that there is an equilibrium with a signaling region equal to ( p , p ). The presence of α − 2 in the denominator of the terms inside the parenthesis of equation (14) indicates that the equilibrium effort, e H ( p ), is big when α is close to 2 (for most of p ∈ ( p , p )). Nevertheless, equation

16

See the proof of Proposition 3.1 for a formal proof that p e H ( p ) is decreasing for small p.

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127

(13) (with V H ( p ) = max{U H , p } and V H ( p ) = p ) implies that e H (·)α −2 is not very large. The following proposition formalizes these intuitions: Proposition 3.6. Let (αn )n be a strictly monotone sequence converging to 2. For each αn , let ( p , pn ) and e H ,n (·) be, respectively, the n signaling region and the effort choice of an equilibrium of the model with α = αn , and assume that p → p and pn → p, for some n

0 < p < p < 1. Then, e H ,n ( p ) → ∞ and e H ,n ( p )αn −2 → 1 for all p ∈ ( p , p ).

When we fix R and let α approach 2 using a sequence (αn )n , the corresponding sequences of e H ,n ( p )αn −2 and V H ,n ( p ), for each p ∈ R, converge to the same limit independently of the direction with which the sequence approaches 2 (the limit of V H ,n is given by (13) with e H (q)α −2 = 1 for all q). The sequence (e H ,n ( p ))n is diverging so, in the limit where α converges to 2, the fixed cost c 0 is increasingly irrelevant (since the signaling time approaches 0). This is consistent with the fact that, when α = 2, equations (11) and (12) admit a solution for V H only when c 0 = 0. 4. Seller’s preferred effort In order to provide some understanding of the previous results, we now analyze the inefficiency caused by the nonobservability of the effort. Instead of studying an alternative model where the effort made by the seller is observable, we provide some technical results and link them to our previous results to have a better understanding of them. The equilibrium analysis of Section 2.1 can be separated into two parts. The first part consists of finding the continuation value of the H -seller for each “conjectured” effort strategy e H (·) if the seller follows it and the buyers believe that the seller follows it. The second part consists of finding a conjecture e H (·) that is incentive compatible for the seller (given that it is unobserved by buyers), that is, which satisfies the FOC (12) (with eˆ H (·) = e H (·)) given V H (·). Since the second part has already been analyzed in Sections 2.1 and 3, we now focus our attention on the first part. For a fixed signaling region R = ( p , p ) and strictly positive e H (·), the continuation value of the H -seller is found using equations (2) and (3) in the maximand of the HJB equation (11) (with eˆ H (·) = e H (·)), which gives us that for all p ∈ R:

0 = −C H e H ( p )α − c 0 + e H ( p )2

p (1 − p )2

σ2

 VH ( p ) + e H ( p )2

p 2 (1 − p )2 2σ2

 VH ( p) ,

(17)

with boundary conditions V H ( p ) = max{ p , U H } and V H ( p ) = max{ p , U H }. The following proposition establishes that when α < 2, exerting a high effort is efficient. Proposition 4.1. Assume

α < 2 and fix a signaling region R = ( p , p ) with 0 < p < p < 1. For each strictly positive policy function

e H (·) ∈ C 1 ( R ), let V H (·, e H ) solve (17) with the corresponding boundary conditions. Then, V H ( p , λ e H ) > V H ( p , e H ) for all λ > 1 and p ∈ R. The intuition for the previous result is as follows. We can obtain an HJB equation equivalent to (17) by dividing it by e H ( p )2 . We see that the two new cost terms, corresponding to the flow and fix costs, are proportional to e H ( p )α −2 and e H ( p )−2 , respectively, and therefore they are decreasing in e H ( p ). Therefore, increasing e H ( p ) does not change the drift and the volatility terms of the new HJB equation and reduces the cost terms (both the fixed and the variable cost, since α < 2). P In the limit where e H (·) → ∞, the cost terms vanish, and V H (·) increases to the expected revenue (E H (·) defined in (15)). When α > 2 the high convexity of the cost function prevents exerting a very high effort from being efficient (notice that when equation (17) is divided by e H ( p )2 , an increase in e H ( p ) still lowers the fixed cost term but now increases the variable cost). Instead, in the absence of a fixed cost, the seller would want to exert a very low effort and let the information about her type be slowly revealed over time.17 Nevertheless, the presence of the fixed cost requires a high flow of information in order to make exerting effort worthwhile. Proposition 4.2. Assume α > 2. There exists a maximal signaling region R + such that, for any signaling region R ⊂ R + , V H (·) as defined in equation (17) is maximal when

 e H ( p ) = e ∗O E ≡

2 c0 C H (α − 2)

 α1

∀p ∈ R .

(18)

Note that the effort that maximizes the payoff of the H -seller is constant. Intuitively, since c H (e ) is convex for e > 0, spreading the effort cost over time is optimal.

17 Moscarini and Smith (2001) find similar results for the optimal effort in an experimentation model, where the type is unknown by the agent who decides the effort. In part, the reason for such similarity is that, given that our seller captures all gains from trade, in the absence of incentive constraints she fully internalizes the gains from the buyers being better informed.

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We can compare the functional form of the effort in our main model (equation (14)) with the case when the effort is perfectly observable (equation (18)). Since K 1 > 0 (see the proof of Proposition 3.4) it is clear that there is a distortion in the effort choice, which makes the equilibrium effort lower than e ∗O E . As a result, as is proven in the proof of Proposition 4.2, R + is strictly bigger than the signaling region of an MSE when α > 2 (see Proposition 3.4), so the seller stops exerting effort too early owing to the unobservability of the effort. Concluding, both when α < 2 and when α > 2, the equilibrium effort is inefficiently low (in the case of α < 2 the H -seller would be better off with any effort higher than the equilibrium effort). This is a consequence of the extra information asymmetry and incentive constraints given by the unobservability of the effort. The seller is unable to fully internalize the gain from increasing the effort, since such a gain (right-hand side of equation (12)) is only a function of the effort choice that buyers believe the seller makes, not the actual effort choice. When α < 2, the inefficiency is more severe for high posteriors (“rush-out effect” when p is low), while when α > 2, it is more severe when p is low (“give-up effect”). As one can expect, in both cases, the distortion in the effort choice generates inefficiently small signaling regions. 5. Conclusions Our analysis sheds light on how endogenous information transmission affects the incentive to transmit information over time. The interaction between the non-observability of the type (idiosyncratic characteristics) and the non-observability of effort (idiosyncratic effort choice) implies that the informativeness of the signal is time-dependent and responds to endogenous incentives. Our model highlights the effect of the cost structure on the way the seller dynamically signals her private information over time. When generating highly informative signals is inefficient, information release is slow, which forces the seller of a high-value firm to stop exerting effort and retain the asset for herself when buyers are pessimistic about its value. In this case, transaction prices will tend to be more dispersed, and the pool of assets that are not traded will be heterogeneous. If, however, highly informative signals are efficient, it is incentive-compatible for the seller of a high-value firm to exert a high (and more efficient) effort when buyers are pessimistic, successfully ensuring that her asset is traded at a high price. So, in this case, prices will tend to be high, and most of the high-value assets will be traded. In the second case, the performance of a firm may not be increasing with its perceived value; after a low sales record firms may increase, in expectation, their sales. Yet, in both cases, we find that the unobservability of the level of effort leads to inefficiently low effort choices and early acceptance decisions by the seller. Future research will be devoted to generalizing our results by, for example, allowing low types to exert effort or introducing additional types. Considering productive signaling, such as productive education, may also introduce new trade-offs, since the uninformed side of the market will value effort more than just as a separation device. Appendix A. Proofs of the results Proof of Lemma 2.1. Proof that A H = A L : The fact that A H = A L is a consequence of the optimality of the H -seller’s strategy and the assumption of no gains from trade from the L-seller. Indeed, notice first that 1 ∈ A H ∩ A L . The reason is that, given that max{U H , W ( p 0 )} ≤  for all p 0 , we have V H ( p 0 ) ≤  for all p 0 , so the No (Unrealized) Deals condition imposes that V H (1) = . Also, given the expression for continuation payoff (7), we have that the expected equilibrium signaling cost for the H -seller (and therefore her expected signaling effort) when p 0 = 1 is 0, and the probability of selling the asset is 1. Then, given that the L-seller can imitate the equilibrium strategy of the H -seller, we have V L (1) = , so W (1) =  and 1 ∈ A H ∩ A L as a result. Hence, if p 0 ∈ (0, 1) is such that p 0 ∈ A L \ A H , we have that, by Buyers’ Zero-Profit, W ( p 0 ) = 0, but rejecting the offer gives the seller a continuation value of , which is a contradiction. Similarly, if p 0 ∈ (0, 1) and p 0 ∈ A H \ A L , then W ( p 0 ) = , and rejecting the offer makes the belief jump down to 0. Rejecting is optimal for the L-seller only if V L (0) = . Nevertheless, since 0 is an absorbing state of the posterior belief process, this is a contradiction. Finally, 0∈ / A L ∪ A H , since if 0 ∈ A L then W (0) = 0 < U θ for all θ ∈ { L , H }, while if 0 ∈ A H \ A L then V L (0) = 0 < U H ≤ W (0), which contradicts the optimality of the L-seller’s strategy. Proof that there is no learning in [0, 1]\ R: We now show that Pr( P t = p 0 ∀t ≤ τθ |e θ , A θ ) = 1 for all θ ∈{ L , H } whenever 1 p0 ∈ / R. Notice that if p 0 ∈ / R then, since A L = A H , either p 0 ∈ A H or p 0 ∈ e − H (0). If p 0 ∈ A H then the result is obvious. If,

1 instead, p 0 ∈ / A H and p 0 ∈ e − H (0), then there is not belief updating in equilibrium, so Pr( P t = p 0 ∀t |e θ , A θ ) = 1. Therefore, the parts 1 to 3 of the statement of the Lemma are immediate. 2

Proof of Proposition 2.1. Assume that an equilibrium with a non-empty signaling region in our base model exists so, in particular, α = 2 (see Section 3 and, specifically, Section 3.4). It is easy to check that Lemma 2.1 also holds when ρ > 0, since the fact that there is no discounting is not crucial in any of the arguments in its proof. As a result, the optimization problem for the seller with discounting can be described as an HJB equation which coincides with (11) replacing the “0” in the left hand side by “ρ ( V˜ H ( p ) − U θ ).” The first order condition (12) remains the same. So, for a given p 0 , the existence of equilibria with a signaling region R containing p 0 relies on the existence of a solution of the following differential equation:

F. Dilmé / Games and Economic Behavior 113 (2019) 116–136





ρ V˜ H ( p )− U θ + c0 +





 α−2 2 

(1 − p ) p

 V˜ H ( p) CH α σ 2



1 −α ( 1 − p ) (1 − p ) p α σ 2

 V˜ H ( p )−

(1 − p )2 p 2 2σ 2

129



 V˜ H ( p) = 0



≡ f ( p , V˜ H ( p ), V˜ H ( p ), V˜ H ( p ))

(19)

such that V˜ H ( p ) ≥ max{U H , p } for all p ∈ ( p , p ) and satisfying the boundary conditions V˜ H ( p ) = max{U H , p } and

 ( p ), V˜  ( p )) defined in the preV˜ H ( p ) = p , where p and p 0 are defined as in (9) and (10). Note that f ( p , V˜ H ( p ), V˜ H H

 ( p) ∈ R  vious equation is continuous in p ∈ (0, 1), V˜ H ( p ) ∈ [U H , ], V˜ H ++ , and V˜ H ( p ) ∈ R. Fix now some equilibrium of our base model with signaling region R = ( p , p ), with 0 < p < p < 1. This implies that there exists a solution of (19) with ρ = 0, denoted V H (·), such that V H ( p ) ≥ max{U H , p } for all p ∈ R and satisfying the boundary conditions V H ( p ) = max{U H , p } and V H ( p ) = p . We consider the case where V H ( p ) > max{U H , p } for all

p ∈ ( p , p ), while the other case (with a weak inequality) is proven analogously.18 Notice first that if εˆ > 0 is small enough,

then there exists an equilibrium, indicated using hats, with signaling region Rˆ = ( pˆ , p − εˆ ), with value function satisfying  (p −ε ˆ ) = V H ( p − εˆ ) < , and where Vˆ H ( p − εˆ ) = ( p − εˆ )  and Vˆ H



pˆ ≡ max p < p − εˆ  Vˆ H ( p )≤ max{U H , p }



is such that pˆ ∈( p , p − εˆ ). Indeed, notice that Vˆ H ( p ) can be written as V H ( p ) +( p − εˆ )  − V H ( p − εˆ ) < V H ( p ), so using that

V H ( p ) > max{U H , p } for all p ∈ ( p , p ) we have that, if εˆ > 0 is small enough, Vˆ H ( p ) > max{U H , p } for all p ∈ ( pˆ , p − εˆ )

and Vˆ H ( p ) < max{U H , p } for all p ∈ ( p , pˆ ) ∪ ( p − εˆ , p ). Notice also that, as εˆ becomes small, pˆ converges to p. For each εˆ > 0, standard ordinary differential equations (ODE) analysis guarantees the existence and uniqueness of the  (p −ε ˆ ) = Vˆ H ( p − εˆ ) (for example, Theorem solution of the equation (19), for any ρ ≥ 0, with V˜ H ( p − εˆ ) = Vˆ H ( p − εˆ ) and V˜ H 20.9 in Olver, 2014). Also, standard ODE analysis (Theorem 20.13 in Olver, 2014) guarantees the continuity of the value function (and the effort function, since the FOCs coincide) with respect to the discount rate ρ . In particular, since Vˆ H ( p ) > max{U H , p } for p ∈ ( pˆ , p − ε ), there exists some ρ¯ > 0 such that if ρ ∈ (0, ρ¯ ) the following function is well defined





p ∗ ( p − εˆ ; ρ ) ≡ max p < p − εˆ  V˜ H ( p )≤ max{U H , p } . Hence, there is an equilibrium in the model with ρ ∈ (0, ρ¯ ) with signaling region ( p ∗ ( p ; ρ ), p ). Again using standard ODE analysis (Theorem 20.13 in Olver, 2014), the continuity of the value function with respect to ρ implies that limρ →0 p ∗ ( p ; ρ ) = p. Finally, for any ε > 0 one can take εˆ small enough that max{| pˆ − p |, εˆ } < ε for ρ < ρ¯ , so d(( pˆ , p − εˆ ), ( p , p )) < ε , and also (by the continuity of the solutions of an ODE with respect to the boundary condition), ρ

sup p ∈ R ∩ R ρ |e H ( p ) − e H ( p )| < ε . The continuity of the value function with respect to the boundary conditions and ρ ensures the reverse result: given any sequence of equilibria in models where ρ → 0 with signaling region converging to some R, there exists an equilibrium when ρ = 0 with signaling region R. 2 Proof of Lemma 3.1. Assume that there is an equilibrium (not necessarily an MSE) with a non-empty interval signaling region ( p , p ), so α = 2 (see Section 3.4). Note that the second derivative of the maximand of (11) with respect to eˆ is

−α (α − 1) C H eˆ αH−2 < 0, so the first order condition is sufficient for optimality in R. Therefore, we impose the equilibrium condition eˆ θ ( p ) ≡ e θ ( p ) in the FOC (12), so we get

α C H e H ( p )α −1 =

p (1 − p ) e H ( p ) V  ( p )

σ

2

⇒ V ( p) =

α σ 2 C H e H ( p )α −2 p (1 − p )

.

(20)

Now, plugging this expression into the maximand of (11) we have a first order differential equation for e H ( p ), given by

 2 0 = −c 0 + α − C H e H ( p )α −1 e H ( p ) + α (1 − p ) p e H ( p ) . 2

(21)

The general solution of this equation is given by (14), where K 1 is a constant to be determined by the boundary conditions on V H (·). Notice that, we can write the following differential equation for V H analogous to (21):

 0 = − c0 − C H

 +

p (1 − p ) V  ( p )

α CH σ2

p (1 − p ) V  ( p )

α CH σ2

 αα−2

 α−2 2

p (1 − p )2  V H ( p) + 2

σ

1 2



 p VH ( p) .

(22)

18 In the alternative case the set Y ≡ { p ∈ ( p , p )| V H ( p ) = max{U H , p }} is not empty. Nevertheless, notice that Y is formed by isolated points (since a linear V H is never a solution of (19) with ρ = 0). Then, the argument of the proof can be applied to each of the open intervals forming ( p, p )\Y .

130

F. Dilmé / Games and Economic Behavior 113 (2019) 116–136

As argued in the proof of Proposition 2.1, the previous equation is well behaved, so its solutions satisfy existence, uniqueness (once some boundary conditions are established) and continuity with respect to the boundary conditions and parameters. Let R now be the signaling region of an MSE, and R˜ ⊂ R the signaling region of another equilibrium denoted using tildes. ˜ and define p, p, p˜ and p˜ as in (9) and Note that if p 0 ∈ / R˜ then V˜ H ( p 0 ) = max{U H , p 0 } ≤ V H ( p 0 ). So, assume that p 0 ∈ R, (10). If p = p˜ and p = p˜ then trivially V H ( p 0 ) = V˜ H ( p 0 ). Otherwise, assume first that p˜ > p and p˜ = p. Then, from the FOC (12) and the form of the policy function (from equation (14)), we have that

p V H ( p) ≡ V H ( p; p, K 1 ) = p  − p

It is easy to verify that

∂ V H ( p ; p , K 1 ) ∂ K1

CH α σ2



K 1 (1 − q)

(1 − q) q

(2 − α ) q

−

 α −2

2 c0

α

dq .

(2 − α ) C H

(23)

< 0.

Therefore, there exists some K˜ 1 > K 1 such that V H ( p˜ ; p , K˜ 1 ) = max{ p˜ , U H } and V˜ H ( p˜ ) = V H ( p˜ ; p , K˜ 1 ). So, given that

< 0, we have that V˜ H ( p 0 ) < V H ( p 0 ). If p˜ ≥ p and p˜ < p we can use an analogous argument, keeping p constant ˜ by writing V H (·) in terms of p instead of p. Note that if R˜ is a strict subset of R then (instead of p) and decreasing p to p, there exist p ∈ R˜ such that V˜ H ( p ) < V H ( p ). 2 ∂ V H ( p; p, K 1 ) ∂ K1

Proof of Lemma 3.2. Assume that an MSE exists and let R be its signaling region. It is clear that V H ( p ) ∈ C 0 ([0, 1], R), since V H ( p ) is continuous in R¯ and V H ( p ) = max{U H , p } is continuous for p ∈ / R. Assume that R is not empty (otherwise the result is trivial). Assume also that p 0 ∈ R, and define p and p as in (9) and (10). Note that by the definition MSE there is no equilibrium with a signaling region containing p or p. We assume V H ( p ) = p  (so p > p − ) and V H ( p ) = p  (the other possible case, when V H ( p ) = U H , is proven similarly). Since V H ( p ) ≥ p  for all p ∈ ( p , p ) and V H (·) ∈ C 1 ( p , p ), we have  ( p ) ≥  and lim  lim p  p V H p  p V H ( p ) ≤ . We need to show that these weak inequalities hold with equality in an MSE. Notice first that, necessarily, p < 1. Indeed, if p = 1 and α < 2, the boundedness of the effort function implies (using  ( p ) = ∞, which is a clear contradiction. If, instead, p = 1 and α > 2, then equation (14) implies that (20)) that lim p →1 V H  ( p ) = ∞. lim p →1 e H ( p ) = ( C 2(αc0−2) )1/α , so we reach a contradiction since, again, lim p →1 V H H   Assume first lim p  p V H ( p ) >  and lim p  p V H ( p ) < . Let V˜ H ( p ) denote the solution of equation (22) which coincides with V H ( p ) for p ∈ ( p , p ) and which, using standard arguments (see Olver, 2014), is defined in an interval ( p − ε , p + ε ),

for some

ε > 0. Notice that, if ε is small enough, V˜ H ( p − ε) < ( p − ε)  and V H ( p + ε) < ( p + ε) . Then, since V˜ H ( p ) + δ

is also a solution of (22) and, if δ > 0 is small enough, V˜ H ( p − ε ) + δ < ( p − ε )  and V H ( p + ε ) + δ < ( p + ε ) , there exist two values p < p and p δ > p such that V H ( p ) + δ = p , V H ( p δ ) + δ = p δ  and V H ( p ) + δ > p  for all p ∈ ( p , p δ ). δ δ δ δ Since the boundary conditions are satisfied, there is an equilibrium with signaling region ( p , p δ ) containing p and p, which δ contradicts the assumption that p and p are not in the signaling region of any equilibrium.  ( p ) >  and V  ( p ) =  (a similar argument can be used when V  ( p ) =  and V  ( p ) < ). Now consider the case V H H H H As before, let V˜ H ( p ) denote the solution of equation (22) which coincides with V H ( p ) for p ∈ ( p , p ) and which is defined

ε > 0. Let Vˆ H be the solution of equation (22) with Vˆ H ( p ) = p  and Vˆ H ( p ) =  − δ , for some small δ > 0. We claim that Vˆ H ( p ) > V˜ H ( p ) for all p ∈ ( p − ε , p ). Indeed, note that V˜ H ( p  ) = Vˆ H ( p  ) for in an interval ( p − ε , p + ε ) for some

 ( p  ) = Vˆ  ( p  ) (with V˜ ( p  ) < Vˆ ( p  )) for some p  < p. Nevertheless, if such a p  exists, then some p  < p only if V˜ H H H H

V˜ H ( p ) − V˜ H ( p  ) + Vˆ H ( p  ) is a solution of equation (22), and its value and its first derivative at p  coincide with that of Vˆ H . Then, Vˆ H ( p ) = V˜ H ( p ) − V˜ H ( p  ) + Vˆ H ( p  ) for all p ≤ p, which contradicts the fact that V˜ H ( p ) = Vˆ H ( p ). So, using the same argument as before, if δ > 0 is small enough there exists some p < p such that Vˆ H ( p ) = max{U H , p } and δ

δ

δ

Vˆ H ( p ) > max{U H , p } for all p ∈ ( p , p ). This, again, contradicts that p is not part of the signaling region of an MSE.

2

δ

Proof of Proposition 3.1. We will prove this proposition by explicitly constructing an MSE. First, notice that the existence of an equilibrium with non-empty signaling region R is trivial. Indeed, take p = UH − ε , for some small ε > 0. Let V H (·; p , v  ) ε

be the solution of equation (22) satisfying V H ( p ; p , v  ) = U H and V H ( p ; p , v  ) = v  . Then, it is clear that ε

ε

ε

ε

ε

ε is small

enough there exists some p ε > UH such that V H ( p ε ; p , /2) = p ε  and such that V H ( p ; p , /2) > max{U H , p } for all ε ε p ∈ ( p , p ε ). ε

Let p be the lowest value p ∈ [ p − , UH ) satisfying that there exists some value v  ∈ R+ and p ∗ > UH such that ∗ V H ( p ∗ ; p , v  ) = p ∗  and V H ( p ; p , v  ) ≥ max{U H , p } for all p ∈ ( p , p ∗ ). By the arguments used in Lemma 3.2, if ∗ ∗ ∗  ( p ; p , 0) =  and V  ( p ; p , 0) = , while V  ( p ; p , v  ) =  if p = p . Notice p > p − then necessarily v  = 0 and V H − ∗ ∗ ∗ H H ∗ ∗ ∗ ∗ ∗  ( p ; p , 0) ≥ 0. that this implies that V H ∗ ∗

F. Dilmé / Games and Economic Behavior 113 (2019) 116–136

131

We prove that, necessarily, p = p − . To see that p > p − is not possible, define ∗

∗

p V H ( p; p) ≡ p  −

p

and k ≡ c 0

 C −H 2/α  2−αα α σ2

2

 p q α −2 (1 − p )2/α (1 − q)−1 α

2 (2 − α )−1 k ( p − q)((1 − p ) p ) 2−α + (1 − q) p

 2−αα dq

(24)

. Equation (24) is found from equation (23) obtaining the value of K 1 corresponding to the boundary

 ( p ) =  (so V ( p ; p ) coincides with V ( p ; p , ) + p  − U ). Notice that, since condition V H ( p ) = p  and V H H H H have that V H ( p ; p ) is defined for all p ∈ (0, 1). Simple algebra shows that

α < 2, we





V H ( p; p) ≥ 0 ⇔

∂ V H ( p; p) ≤0 ∂p α

⇔ k − (α (1 − p ) − 1) ((1 − p ) p ) α−2 ≥ 0 .   

(25)

≡ g ( p)

Since

α<

2 then the equation g  ( p )

= 0 has no solution, lim p→0 g ( p ) = −∞ and lim p →1 g ( p ) = +∞, and therefore 





there exists a unique p †† ∈ (0, 1) such that V H ( p ; p ) > 0 for p > p †† , V H ( p †† ; p †† ) = 0 and V H ( p ; p ) < 0 for p < p †† . 

††

††

As a result, necessarily it is the case that, since V H ( p ; p ) ≥ 0, p ∗ ≥ p . Furthermore, from equation (25) we have that lim p →0 V H ( p ; p ) = −∞. 



We now show that, whenever V H ( p ; p ) > 0, we have that V H ( p ; p  ) is increasing and V H ( p ; p  ) is decreasing in p  in a 

neighborhood of p for all p ∈ (0, p ). To see this, take p such that V H ( p ; p ) > 0, and 



ε > 0 such that V H ( p + ε; p + ε) > 0,

V H ( p ; p + ε ) <  and V H ( p ; p + ε ) > p , which exists by continuity of the solutions of equation (22) with respect to its boundary conditions. Assume that there exists some p  < p such that V H ( p  ; p ) > V H ( p  ; p + ε ). This implies that,   necessarily, there is some p  < p such that V H ( p  ; p ) = V H ( p  ; p + ε ). In this case, we have that, for all p, we have

V H ( p ; p ) = V H ( p ; p + ε ) + V H ( p  ; p ) − V H ( p  ; p + ε ) . Indeed, notice that only the first and second derivatives of V H appear in (22) (but not V H ( p ) directly), so if V H (·) is a   solution then V H (·) + C is a solution for all constants C ∈ R. This is a contradiction, since V H ( p ; p ) =  > V H ( p ; p + ε ). Therefore, if p > p − , we have that since V H ( p ; p ) is increasing in p for p > p †† , there is some ε > 0 and p ∈ ( p − , p ) ∗ ε ∗ such that V H ( p ; p + ε ) = U H and V H ( p ; p + ε ) ≥ max{U H , p } for all p ∈ ( p , p + ε ), which contradicts the definition of ε

ε

p .19 As a result, the only candidate for MSE is ( p , p ∗ ) defined above, which satisfies p = p − . ∗

∗

∗

Assume that there is another equilibrium with signaling region ( p  , p  ), with p  > p ∗ . Using an argument analogous to

the one in Lemma 3.2 there must exist some equilibrium with signaling region equal to ( p  , p  ) satisfying the smooth pasting conditions, for some p  > p ∗ . Still, this is impossible given that V H ( p ; p ) is increasing for p > p ∗ (since p ≥ p †† ), which implies that V H ( p ; p  ) > max{U H , p } for all p ∈ [ p − , p  ). We finally prove that the equilibrium effort in an MSE, e H (·), is decreasing for all p ∈ R, and that p e H ( p ) is decreasing 0 if p is small enough. Since C 2(2c− α ) > 0, it must be the case that K 1 > 0 (in equation (14)) in order for the term inside H the power function to be non-negative. This makes e H (·) clearly decreasing. Notice also that, for a fixed K 1 > 0, p e H ( p ) is increasing when p is small:

d dp



p e H ( p) =

(2 c 0 + C H K 1 ) e H ( p )1−α (2 − α ) C H p



K 1 (α − 1)

α (2 c0 + C H K 1 )

Given that lim p → p e H ( p ) ≥ 0 we have that K 1 ≥ > 0. 2

2 c0 p . C H (1− p )

 −p .

−1 This implies that when p < αα p we have that

d dp





p e H ( p)

Proof of Proposition 3.2. The value function for an MSE with R = ( p − , p ∗ ( p − )) is given by V H (·; p ∗ ( p − ), k) in (24), where we explicitly write k as an argument. Note that when α < 2, k ≡ k(σ ) (defined in the proof of Lemma 3.2) is decreasing in σ . Therefore, if σ increases, the integrand of the expression (24) increases for each given q (keeping p the same). So, since σ1 < σ2 , V H ( p − ; p ∗ (σ1 ), k(σ2 )) < V H ( p − ; p ∗ (σ1 ), k(σ1 )) = U H , where p ∗ (σ ) is the value of p ∗ for the parameter value σ . Hence, since V H ( p − ; p , k(σ1 )) is increasing in p, we have that p ∗ (σ1 ) < p ∗ (σ2 ). 2 Proof of Proposition 3.3. We first prove that lim p − 0 p ∗ ( p − ) < 1. To do this, note that, for a fixed p, from the first-order  ( p ) = , we can obtain the value of K as a function of p as follows: condition (12) and the smooth-pasting condition V H 1

19



Notice that V H ( p ; p + ε ) = V H ( p ; p , V H ( p ; p + ε )), so p is not minimal satisfying the conditions above. ε

ε

∗

132

F. Dilmé / Games and Economic Behavior 113 (2019) 116–136



K 1( p) =

α

c 0 p 2 k + (2 − α ) ((1 − p ) p ) α−2



(1 − p ) C H k

(26)

,

where k is defined in the proof of Lemma 3.2. Using this, and the HJB equation (11), we have that





V ( p) =

α

2  k ((1 − p ) p ) 2−α − α (1 − p ) + 1



α p (1 − p )

.

 ( p ) is continuous and Note that the right hand side of the previous equation tends to infinity when p → 1. Since V H  ( p ) < 0 for some p < p, which is a contradiction. V  ( p ) = , we have that if p is large enough V H We now prove that lim p − 0 e H ( p ) <∞ for all p ∈(0, p ∗ (0)). To see this, notice from (26) that since lim p − 0 p ∗ ( p − ) < 1 then, by equation (26), lim p − 0 K 1 ( p ∗ ( p − )) <∞. Therefore, given our expression for the equilibrium effort (14), the result holds. Next, we show that the belief process exits the signaling region R of an MSE in finite time. So, assume p 0 ∈ R and define p and p as in (9) and (10). To see this, fix some ρ > 0 interpreted as a fictitious discount rate, and define B θ ( p 0 ) as the expected discounting of the time when the belief process exits R conditional on the type being θ ∈ { L , H }. More formally, B θ ( p 0 ) ≡ E[e −ρτ R |e θ , p 0 ], where τ R is the stopping time given by the first time the price process is either p or p. Notice that the limit of B θ ( p 0 ) as ρ tends to 0 gives the probability that the process exits R. Then, B θ ( p ) satisfies the following equation for all p ∈ R:

ρ B θ ( p ) = μ˜ (eθ ( p ), p , e H ( p )) B θ ( p ) + 12 σ˜ ( p , e H ( p ))2 B θ ( p ) , ˜ (e θ ( p ), p , e H ( p )) = 0 for with boundary conditions B θ ( p ) = B θ ( p ) = 1. Since 0 < p < p < 1; and B θ ( p ) ≥ 0, e H ( p ) > 0 and μ all p ∈ ( p , p ) and θ ∈ { L , H }; as ρ → 0 the solution of this differential equation converges to B θ ( p ) = 1 for all p ∈ ( p , p ) and θ ∈ { L , H }. Finally, we prove that lim p − 0 E[τ L | P τL = p − , p 0 = p ] <∞ for all p ∈(0, p ∗ (0)). To do prove this, note that

 e H ( p) ≥

K 1 (1 − p )

1/α

(2 − α ) p



K 1 (1−p ) (2−α ) p



−

2 c0 C H ( 2 −α )

K 1 (1 − p ) ( 2 −α ) p

1/α

1/α

≥ K 1∗



1− p p

1/α

≡ e ∗H ( p ) ,

(27)

where K 1∗ is obtained using equation (26), verifying that the right hand side of the first equality is increasing in p, and then evaluating the expression at p = UH (since p ∗ ( p − ) ≥ UH for all p − > 0). Let T θ ( p ) be the expected time before an offer is accepted when the effort exerted by the seller is e θ , for θ ∈ { L , H }, that is



T θ ( p0 ) ≡ E





 τθ

τθ  p 0 , et = eθ ( P t ) = E

0





ds  p 0 , et = e θ ( P t ) .

Therefore, T θ (·) can be thought of as the value function for a flow payoff of 1 while the belief process is in R and 0 once it exits it. Hence, T θ (·) satisfies the following HJB equation:

˜ (e θ ( p ), p , e H ( p )) T θ ( p ) + 0=1+μ

1 2

σ˜ ( p , e H ( p ))2 T θ ( p ) ,

with boundary conditions T θ ( p − ) = T θ ( p ∗ ( p − )) = 0. Given that providing less effort generates a slower belief updating, one can obtain an upper bound on the expected time by using e ∗H (defined in equation (27)) instead of the equilibrium effort. After some amount of algebra, T L ( p ) can be bounded in the following way





α 2 σ 2 (1 − p ) h( p ) ( p − p ) + (1 − p ) h( p ) ( p − p ) − (1 − p ) ( p − p ) h( p ) T L ( p )≤ (2 − α ) K 1∗2 (1 − p ) ( p − p )



α 2 σ 2 (1 − p ) h( p ) p − (1 − p ) p h( p ) , ≤ (2 − α ) K 1∗2 (1 − p ) p where h( p ) ≡



T L ( p) =

p  α 1− p

2−α

and where, to keep notation short, we write p instead of p ∗ ( p − ). Since we can also write

( p − p )(1 − p ) (1 − p )( p − p )   

E[τ L | P τL = p − ] +

(1 − p )( p − p ) (1 − p )( p − p )

E[τ L | P τL = p ] ,

≥ ((1p−−pp))p

and since p ∗ ( p − ) is bounded away from 1, E[τ L | P τL = p − ] is uniformly bounded for all p − . Proof of Proposition 3.4. In this proof we will use V H defined in (24). Notice that, for p ∈ ( p ( p ), 1) with 0

2

α > 2, V ( p ; p ) is well defined for all

F. Dilmé / Games and Economic Behavior 113 (2019) 116–136

 p ( p ) ≡ max 0, p − 0

133



1− p 2 k ( p (1 − p ))α /(2−α ) p (α − 2) − 1

(28)

.

Also notice that, if p ( p ) ≥ 0, lim p  p ( p ) V ( p ; p ) = 0. Furthermore, now equation g  ( p ) = 0 defined in equation (25) has 0 0 two solutions, among which

p† ≡

1

2+



(29)

α −2 α −1

is the only minimum of g. In this case, since g (0) = g (1) = k > 0, g has at most two zeros, and in case there are two, †

†

†

†

†

†

denoted p 1 and p 2 , we have 0 < p 1 < p † < p 2 < 1, and g ( p ) ≥ 0 if and only if p ∈ [0, p 1 ] ∪ [ p 2 , 1]. 

Recall that, as we show in the proof of Proposition 3.1, whenever p is such that V H ( p ; p ) > 0, V H ( p ; p  ) > 0 is locally 

increasing in p  and V H ( p ; p  ) is locally decreasing in p  in a neighborhood of p, for all p ∈ ( p ( p ), p ). So, we have 2 cases: 0

1. The first is that g has either no zero or one zero, so g ( p ) ≥ 0 for all p ∈ (0, 1), and g ( p ) = 0 only if p = p † . In this case notice that, by the previous result, V H ( p ; p ) is increasing with respect to p for all p ∈ (0, 1) and p ∈ ( p ( p ), p ), and 0



from equation (24) we have lim p →1 V H ( p ; p ) = . Also, notice that V H ( p ; p ) <  for all p ∈ ( p ( p ), p ).20 Hence, there 0 is a unique p ∗ such that either V H ( p − ; p ∗ ) = U H (in this case p ≡ p − ), or V H ( p ( p ∗ ); p ∗ ) = U H (so p = p ( p ∗ )). As ∗ 0 ∗ 0 a result the unique MSE has an interval signaling region, R = ( p , p ∗ ). Since k is increasing in c 0 and g is increasing in ∗ k, this happens whenever c 0 ≥ c¯ 0 for some c¯ 0 > 0. †

†

2. The second case is when two zeros of g exist (p 1 and p 2 ), so c 0 < c¯ 0 .

†

†

We first prove that, in this case, there exist two unique values p > 0 and p 1 < 1, with p ≤ p 1 < p 2 ≤ p 1 , such that 1

1



†

†

V H ( p ; p ) = V H ( p ; p 1 ) > p  for all p ∈ ( p , p 1 ). To see this, recall first that V ( p ; p ) < 0 for all p ∈ ( p 1 , p 2 ) and, in 1

1



particular, V H ( p † ; p † ) < 0. So, V H (·; p † ) + ε , for

ε > 0 small enough, is such that there exist two values p ε and p ε , with p < p < p ε , so that V H ( p ; p ) = p  and V H ( p ε ; p † ) = p ε , and V H ( p ; p † ) > p  for all p ∈ ( p , p ε ). Let ε¯ ε ε ε ε †

†



be the highest value with this property. It is clear that either V H ( p ε¯ ; p † ) =  or V ( p ¯ ; p † ) = . Assume the first (the ε



†

other case is analogous). Since V H (·; p ε¯ ) = V H (·; p † ) + ε and V H ( p ε¯ ; p ε¯ ) ≥ 0, we have p ε¯ ≥ p 1 . So, since V H ( p ; p ) is increasing in p for p ≥ 

† p1

and lim p →1 V H ( p ; p ) =  for all p, there exist some p and p 1 such that V H ( p , p 1 ) = p  1

1

1

and V H ( p , p 1 ) = . Figs. 5(a) and 5(b), in gray line (except in the regions where it coincides with the equilibrium 1

continuation value) V (·, p 1 ) (passing through the points ( p , p ) and ( p 1 , p 1 ) with slope ). We can then divide 1 1 the argument in two sub-cases: (a) Assume first that V H ( p ( p 1 ); p 1 ) ≤ U H . Let p ∗ ∈ [ p 1 , 1) be highest for which either V H ( p ( p ∗ ); p ∗ ) = U H (so p = 0

∗

0

p ( p ∗ )) or V H ( p − ; p ∗ ) = U H (so p = p − ). It is then clear that ( p , p ∗ ) is the signaling region of an MSE. Notice 0 ∗ ∗ † † that, from equation (25), we have p 1 → 0 and p 2 → 1 as k → 0. So, since k is increasing in c 0 , this case holds when c 0 ≤ c˜ 0 , for some c˜ 0 . This is case is depicted in Fig. 5(a). (b) Assume now, instead, that V H ( p ( p 1 ); p 1 ) > U H . Let then p 2 be the highest value such that either V H ( p ( p 2 ); p 2 ) = 0

0

U H (in which case p ≡ p ( p 2 )) or V H ( p − ; p 2 ) = U H (in which case p ≡ p − ). It exists because V H ( p ( p ); p ) < U H 2 0 2 0

for p = UH + ε , for ε > 0 small enough. Notice that, necessarily, p 2 ∈ ( UH , p ) ∪ ( p 1 , 1). Furthermore, since V H ( p ; p ) 1  is increasing and V H ( p , p ) is decreasing in p for p ∈ ( UH , p ), we have that V H ( p ; p ) > U H for all p ∈ ( p ( p ), p ) 1

0

and p ∈ ( p 1 , 1). So, necessarily, p 2 ∈ ( UH , p ). 1 In this case, the signaling region of an MSE is ( p , p 2 ) ∪ ( p , p 1 ). The corresponding equilibrium value function is 2 1 depicted in Fig. 5(b). 2 Proof of Proposition 3.5. See the proofs of Propositions 3.1 and 3.4.

2

Proof of Proposition 3.6. Let’s first consider a strictly decreasing sequence (αn )n converging to 2 (so αn > 2 for all n ∈ N) and a sequence of signaling regions ( p , pn )n converging to some ( p , p ). Fix any p ∈ ( p , p ), so p ∈ ( p , pn ) for all n high n n enough. Hence, there exists a sequence ( K 1,n )n so that equation (14) can be written as write

e H ,n ( p )αn −2 =

20



2 c0

(αn − 2) C H

−

(1 − p ) K 1,n (αn − 2) p

(αn −2)/αn (30)

.





Indeed, notice that if p  is such that V H ( p  ; p ) =  we have V H ( p ; p ) = V H ( p ; p  ) for all p ∈ ( p ( p ), p ). Since V H ( p  ; p  ) > 0, p  is a minimum of

V H ( p ; p ) − p , so V H ( p ; p ) − p 

has no local maxima. So, the only p 

∈ ( p 0 ( p ), p )

0

satisfying V H ( p  ; p ) =  is p 

= p.

134

F. Dilmé / Games and Economic Behavior 113 (2019) 116–136

Fig. 5. We plot (in black) the MSE’s value function V H for different parameter values. In (a), it coincides with V H (·; p ∗ ) in ( p , p ∗ ) (in dotted lines). We ∗ also plot V H (·; p 1 ) (in gray). In (b) V H coincides V H (·; p 1 ) in ( p , p 1 ) (in gray) and with V H (·; p 2 ) in ( p , p 2 ) (in dotted lines). 1

2

Notice that, since the term inside the parenthesis is positive. The term on the right hand side of the equation does not converge to 1 only if K 1,n tends to −∞ and is such that lim supn→∞ (− K 1,n /(αn − 2))(αn −2)/αn > 1. Consider now a strictly increasing sequence (αn )n converging to 2 (so αn < 2 for all n ∈ N) and a sequence ( p , pn )n n converging to some ( p , p ). The limit of the right hand side of expression (30) does not tend to 1 only if K 1,n tends to +∞

and is such that lim supn→∞ ( K 1,n /(αn − 2))(αn −2)/αn < 1. Let η denote lim supn→∞ e H ,n ( p )(αn −2)/αn ∈ R+ ∪ {+∞}, and notice that it is independent of p ∈ (0, 1). Notice also that, as we showed, η ≥ 1 when (αn )n is strictly decreasing and η ≤ 1 when (αn )n is strictly increasing. Furthermore, V H ,n (·) by equation (13), we have that

p lim sup V H ,n ( p ) = max{U H , p } + n→∞

(1 − q) q

n

p

CH 2σ2

η dq

n

and V H ,n ( p ) and V H ,n ( pn ) converge to max{U H , p } and p , respectively. Therefore, necessarily the superior limit of n

e H ,n ( p )α˜ n −2 has to be independent of the monotonicity of (αn )n , which implies that such a limit is equal to η = 1. In both cases, the fact that the term inside the parenthesis of the right hand side of (30) is positive implies that e H ,n → ∞. 2 Proof of Proposition 4.1. Fix p 0 ∈ R and define p and p as in (9) and (10). Fix e H ∈ C 1 ( p , p ) positive. The expected payoff of the H -seller from exerting effort e H in a signaling region ( p , p ) is given by the following HJB equation





( p − 1)2 p p V H ( p ) + 2 V H ( p ) 0 = −c 0 − C H e H ( p )α + e H ( p )2 , 2σ2

(31)

and boundary conditions V H ( p ) = max{U H , p } and V H ( p ) = p . Assume first c 0 = 0. Fix some function e H (·) strictly positive in ( p , p ) and let V H ( p ; e H ) be the corresponding solution of (31). Consider the following decomposition: V H ( p ; e H ) ≡ n V h ( p ) + V r ( p ; e H ), where V h is the solution of the homogeneous equation (solving p V H ( p ) + 2 V Hn ( p ) = 0) satisfying the n n r boundary conditions V H ( p ) = max{U H , p } and V H ( p ) = p , and V H being the reminder. n Notice that V H is the expected payoff net of signaling costs (note that the homogeneous equation is “as if” C H = 0), which coincides with the expected accepted price offer or value of retaining the asset conditional on being type H (notice r that it is given in equation (15)). Alternatively, note that V H ( p ; e H ) satisfies (31) and V Hr ( p ; e H ) = V Hr ( p ; e H ) = 0. Clearly, r V H ( p ; e H ) < 0 for all p ∈ ( p , p ), since it is the solution of a boundary problem with negative flow payoff and with 0-value at the boundary. Then, it is the case that for all λ > 1 we have r VH ( p ; λ e H ) = λ2−α V Hr ( p ; e H ) > V Hr ( p ; e H ) . r The first equation is owed to the homogeneity of equation (31), both λα −2 V H ( p ; e H ) and V Hr ( p ; λ e H ) satisfy the same equations and boundary conditions (equal to 0 at the boundary). Since α > 2, increasing the effort by a factor λ > 1, the r absolute value of V H ( p ; λ e H ) is reduced by a factor λ2−α < 1. n r remains unchanged and V H decreases in absolute Therefore, increasing the effort by the factor increases V H , since V H n value. As λ → ∞, V H converges to V H , that is, signaling waste asymptotically disappears. When a term −c 0 on the right hand side of equation (31) a similar argument applies: in this case, a high effort speeds up the time it takes for the posterior to leave ( p , p ) and, as a result, decreases the signaling waste derived from the fixed cost of signaling. 2

F. Dilmé / Games and Economic Behavior 113 (2019) 116–136

135

Proof of Proposition 4.2. The problem of maximizing the value function of the H -seller can be written as a regular stochastic control problem, since now there is no incentive constraint:

 0 = max

e H ( p)





( p − 1)2 p p V H ( p ) + 2 V H ( p ) − c 0 − C H e H ( p )α + e H ( p )2



2σ2

The First Order Condition of the previous equation is



.



( p − 1)2 p p V H ( p ) + 2 V H ( p ) 0 = −α C H e H ( p )α −1 + 2 e H ( p) . 2 2σ

Note that since α > 1 the Second Order Condition is satisfied. Using the previous two equations to solve for e H ( p ) it is easy to verify that the statement of the proposition is true (note that the terms of both equations involving p are identical). The region R + is found using the smooth pasting conditions. Since e ∗O E is strictly higher than our equilibrium effort and, for a signaling region, the continuation value of the seller is strictly higher when the effort is e ∗O E , if R is the region of an MSE, R is a strict subset of R + . 2 Appendix B. Other derivations Before deriving equation (1), we show that it has a unique solution. To see this, fix a strategy profile ( A θ , e θ )θ∈{ L , H } and 1 let Z H ≡ e − / R = AL ∪ AH ∪ Z H . H (0) be the set of posteriors where the effort of the H -seller is 0. Assume first that p 0 ∈ In this case, buyers learn from the public signal, which they perceive as informative because the effort they believe that the H -seller exerts is positive. The Girsanov’s Theorem ensures the probability measures over the paths of two diffusion processes with the same volatility but different bounded drifts are equivalent, that is, they have the same zero-probability events and, as a result, the Bayes’ rule is used to update P t from P 0 = p 0 , which we show below that gives equation (1). ˆ ∪ Z H ), the drift and the volatility of P are Lipschitz continuous Since eˆ and e H are Lipschitz continuous in [0, 1]\( A L ∪ A H ∪ A ˆ (with respect to P ). Also, if P t ∈ Z H \( A L ∪ A H ∪ A ) then dP t = 0, so there is no learning after t in equilibrium and P s = P t for all s > t. Hence, the stochastic differential equation (1) has a unique (strong) solution with continuous paths ( P t )t ≤τL ∧τ H ∧τˆ . Derivation of equation (1) We mimic the proof of Proposition 1 (p. 787) in Faingold and Sannikov (2011) to derive equation (1). In order to make the argument clear, we keep e L in the formulae below (and finally impose e L ≡ 0). Fix a belief process ( P t )t measurable with respect to the filtration F B . Fix also a strategy profile (e θ , A θ )θ∈{ L , H } and an effort-choice strategy eˆ . The strategy of each type of the seller induces a probability measure over the paths of the public signal ( X t )t ≥0 . From Girsanov’s Theorem we can find the ratio ξt between the likelihood that a path ( X s )ts=0 such that Xs ∈ / A L ∪ A H for all s < t arises for type H and the likelihood that it arises for type L. When the actual effort-choice strategy is eˆ , this ratio is characterized by

dξt = ξt σ −1 (e H ( P t ) − e L ( P t )) d Bˆ t , ξ0 = where Bˆ t ≡

t 0

p0 1 − p0

(32)

,

σ −1 (d X s − eˆ ( P s ) ds) is a Brownian motion under the probability measure generated by the strategy eˆ . By the

Bayes’ rule, the posterior P˜ t after observing a path ( X s )ts=0 is

P˜ t =

p 0 ξt

(33)

.

p 0 ξt + 1 − p 0

By Ito’s formula,

p 0 (1 − p 0 )

d P˜ t =

dξt −

2 p 20 (1 − p 0 )

ξt2 (e H ( P t ) − e L ( P t ))2

dξt 2 ( p 0 ξt + (1 − p 0 ))2 ( p 0 ξt + (1 − p 0 ))3 = P˜ t (1 − P˜ t ) (e H ( P t ) − e L ( P t )) d Bˆ t − P˜ t2 (1 − P˜ t ) (e H ( P t ) − e L ( P t ))2 dt  = P˜ t (1 − P˜ t ) (e H ( P t ) − e L ( P t )) σ −2 d X t − ( P˜ t e H ( P t ) + (1 − P˜ t ) e H ( P t )) dt .

Since d X t = eˆ ( P t ) dt + σ dB t and e L ( P t ) = 0 for all P t , we have that P t = P˜ t almost surely if and only if equation (1) holds. Conversely, suppose that ( P t )t ≥0 solves equation (1) with initial condition P 0 = p 0 . Define ξt using expression (33), i.e.

ξt =

1 − p0

Pt

p0

1 − Pt

.

Then, applying Ito’s formula to the expression above gives equation (32), hence ξt must equal the ratio between the likelihood that a path ( X s )ts=0 arises for type H and the likelihood that it arises for type L. Thus, P t is determined by Bayes’ rule and the belief process is consistent with E H ,t and E L ,t = 0.

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F. Dilmé / Games and Economic Behavior 113 (2019) 116–136

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