Bank of Israel Research Department

Boom-Bust Cycles of Learning,

Investment and Disagreement

Osnat Zohar*

Discussion Paper 2019.06

May 2019

_______________________

Bank of Israel - http://www.boi.org.il

* Bank of Israel and the Hebrew University of Jerusalem.

E-mail: osnat.zohar@boi.org.il, Phone: 972-2-6552652

Any views expressed in the Discussion Paper Series are those of the authors

and do not necessarily reflect those of the Bank of Israel

91007ירושלים 780חטיבת המחקר, בנק ישראל ת"ד

Research Department, Bank of Israel. POB 780, 91007 Jerusalem, Israel

1

מחזורים אסימטריים של למידה, השקעה וחוסר הסכמה

אסנת זהר

תמצית

צומחות בהדרגה ומדי פעם , כלומר, הן אסימטרית בדינמיקה קרובות לעתים מאופיינות לעתיד ציפיות וכן ריאלית פעילות

השקעה זה במודל. למידע פעילות בין חזק קשר ישנו כאשר מופיעה זה מסוג דינמיקה כי מראה המאמר. מתכווצות בעוצמה

מייצר המשקל שיווי. מפעולותיו רק שני שחקן שברשות המידע על לומד אחד ושחקן, העולם מצב על פרטי מידע חושפת פעילה

הפעילות התאוששות ולאחריה סימולטנית להיות נוטה ממנו היציאה; הדרגתית היא לשוק הכניסה: אסימטריים מחזורים

הראשון השחקן של במידע צופה השני השחקן אם. ציבורי הוא המידע כאשר מתעצם במחזור הסימטריה חוסר. איטית היא

מתאם מראות המודל של סימולציות, לבסוף. הבסיסי למודל בהשוואה לשוק כניסתו את לדחות נוטה הוא, בפעולותיו רק ולא

.במודל הלמידה למנגנון רבה במידה מיוחס זה ומתאם, האמונות פיזור לבין בשוק ההשקעה רמת בין חיובי

Boom-Bust Cycles of Learning, Investment and Disagreement

Osnat Zohar∗

May 27, 2019

Abstract

Real activity as well as expectations often exhibit asymmetric dynamics, namely, theyincrease gradually with occasional large downturns. Such dynamics emerge in a modelwith strong feedback between activity and information. In the model, active investmentreveals private information about the state of the world. An agent (Follower) only learnsabout another agent’s (Loner’s) signals from his actions. Equilibrium in the model generatesasymmetric cycles: Entry to the market is gradual; exits tend to be abrupt and are followedby slow recoveries. The asymmetry in the cycle is magnified when information is public. IfFollower observes Loner’s payoffs and not just his actions, he is more likely to defer his entrycompared to the benchmark model. Finally, model simulations show a positive correlationbetween investment and dispersion of beliefs which is largely attributed to the learningmechanism in the model.

Keywords: dynamic learning, asymmetric cycles, slow recovery, private information.JEL Classification: C73, D82, D83, E32.

1 Introduction

Real activity and expectations about future conditions exhibit asymmetric dynamics, namely,they usually increase gradually with occasional large downturns. This is reflected, inter alia, innegative skewness of indicators from business surveys. The negative skew is evident in indicatorsof both past and expected activity, as well as across different countries and industries (Table1).1 This paper proposes a mechanism that explains such asymmetric dynamics, driven by thefeedback between real activity and information.

The mechanism is based on the observation that while rational agents rely on availableinformation to choose optimal actions, these actions may also generate information. Specifically,this is true in situations where active investment provides excess information regarding potentialprofitability. For example, a venture capital fund will learn more about the technology underdevelopment and about the business plans of a startup once the fund invests in it; a firm willlearn more about potential revenue at a new geographic location after it opens a new branch inthe area. While investment is a source for potential monetary gains in these settings, it is alsoa tool for agents to acquire information regarding the asset in which they are investing. Theadditional role of investment increases agents’ incentives to enter the market. However, if agentsare able to observe decisions by other rational agents, the incentive to learn by investment ismoderated as agents “free-ride” on the information acquired by others.

∗Bank of Israel and Hebrew University of Jeruslaem, osnat.zohar@boi.org.il. This paper is part of my doctoraldissertation at the Hebrew University of Jeruslaem. I would like to thank my supervisor Ilan Kremer for hisexceptional guidence. I would also like to thank Marina Halac, Motty Perry and Assaf Patir for their helpfulcomments. The views in this paper do not necessarily reflect the views of the Bank of Israel.

1Admittedly, the skewness in the construction sector is not significantly different from zero, both for the UKand the Euro area. This could be due to unique features of the housing market that affect the cyclical patternsobserved in it [Burnside et al. (2016)].

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Table 1: Skewness of Activity Indicators Compiled from Business Tendency Surveys (1985M01-2018M12)

Sector Country Skewness of Indicator ObservationsPast Tendency Expected Future

Tendency

Manufacturing Euro area -1.23* -1.48* 408UK -0.86* -1.01* 408USA -0.89* - 408

Construction Euro area -0.24 - 408UK 0.02 - 408

Retail Trade Euro area -0.09 -0.77* 408UK -0.45* -0.99* 408

Services Euro area -0.45* -0.13 285UK -0.71* -0.77* 264

Non-Manufacturing USA -1.29* - 258

Source: OECD - Business Tendency and Consumer Opinion Surveys (MEI).

Notes: Indicators are seasonally adjusted balances of answers (the difference between the percentages ofrespondents giving positive and negative replies) regarding past and future tendencies of production (for themanufacturing sector), activity (for construction and retail trade) or demand evolution (for services). Theyare available at a monthly frequency for the period 1985-2018. Asterisks represent skewness that is larger,in absolute terms, than 2

√V S where V S = 6n(n−1)

(n−2)(n+1)(n+3)is an estimator of the variance of skewness and

n is number of observations.

To highlight the mechanisms in play I focus on a setting of two agents, Loner and Follower,where both can invest in a similar asset. When choosing to invest, an agent receives a privatelyobserved payoff that, conditional on the state of the world, is drawn independently of the otheragent’s payoff. The two agents differ in the ability to observe each other’s actions. Loner isassumed to be “isolated” - he cannot observe the actions of Follower. However, Follower canobserve Loner’s investment decisions through which he can learn about Loner’s private signals.

Equilibrium in the model exhibits two main features. First, asymmetric dynamics of beliefsand investment. While agents’ entry to the market is gradual, the exit is usually abrupt. Thebasic mechanism that generates these results is as follows. Agents enter the market when theybecome optimistic enough regarding the state of the world (once their belief crosses a certainthreshold). Entry is gradual, as Follower observes Loner’s actions and thus learns about thestate without engaging in active investment himself. This means that Follower prefers to “free-ride” on Loner’s information and defer his entry. Once agents enter the market they startreceiving payoffs which provide private information about the state of the world. If an agentreceives bad news he opts to exit the market. In a situation where Loner receives a negativesignal before Follower and is seen leaving the market, Follower experiences a “Wile E. Coyotemoment”.2 Like the cartoon character who runs off the cliff in pursuit of the Road Runner,only to find himself hovering in mid-air before crashing to the ground, Follower realizes that hisoptimism is groundless. He immediately exits the market, causing an abrupt crash in activity.In these cases, which are shown to be more likely, the “boom” in activity is more gradual thanthe bust.

The second feature of equilibrium in the model is slow recovery, meaning that after activitycrashes it takes a long time for agents to resume investment. This result follows from thetight connection between activity and information. The crash in activity cuts off the flow ofinformation as agents no longer observe the signals generated by active investment. Specifically,

2The phrase was coined by Krugman (2007) in reference to foreign investors financing what he viewed as anunsustainable U.S. current account deficit in the mid 2000s.

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they cannot observe improvements in the state of the world so they wait until enough time haselapsed since receiving the adverse signal before reentering the market.

These asymmetries are magnified in a public information setting, i.e., if Follower observesLoner’s payoffs and not just his actions. While equilibrium in the public information settingexhibits the same qualitative properties (gradual entry due to free-riding, abrupt exits andprolonged periods on inactivity), it generates patterns with amplified asymmetry. Directlyobserving Loner’s signal makes it less costly for Follower to enter the market, compared to thebaseline setting in which he observes Loner’s actions. Follower’s threshold for entry is thuslower in the public information setting. Nonetheless, in equilibrium he is more likely to enterthe market later. The intuition is that Follower can now tell if Loner is staying in the marketbecause he received good news or because he has not yet received bad news. In the privateinformation setting these two alternatives are indistinguishable so Follower quickly becomes veryoptimistic when he observes Loner stay in the market, and so his belief crosses his thresholdearlier. This means that public information generally defers entry to the market and amplifiesthe asymmetry of the cycle.

The final part of the paper demonstrates that disagreement, i.e., dispersion of beliefs, is procyclical: it is positively correlated with total investment. This is because when both agentsare in the market, information is based more on private signals, making beliefs more dispersed.Once activity crashes, beliefs become perfectly aligned and remain that way until investmentis resumed. The learning mechanism in the model plays an important role in generating thispositive correlation and absent of it disagreement is less correlated with activity and may evenbe counter cyclical.

The paper is closely related to the literature on social learning in which agents’ actionsgenerate information (see Horner and Skrzypacz (2016) for a recent survey). One class of modelsin this literature is that of “strategic bandit models” where the signals generated by agents’actions are publicly observed [Bolton and Harris (1999); Keller et al. (2005); Keller and Rady(2015)]. In another class of models signals are private, however, the action that generates themis irreversible [Chamley and Gale (1994); Caplin and Leahy (1994); Rosenberg et al. (2007);Moscarini and Squintani (2010); Murto and Valimaki (2011, 2013)]. Frick and Ishii (2015)construct a model with publicly observed signals and irreversible actions. My model differsfrom this strand of literature in two aspects. First, these models assume that the state of theworld is constant while I allow for the state to change over time in order to study the dynamicsof the main variables. Specifically, in my model the state of the world follows a Markov chainprocess. Second, my model is the first, to the best of my knowledge, to tackle a setting of privateinformation with reversible actions (agents are free to enter and exit the market in any period).The focus on reversible actions makes it possible to explore the dynamics of investment andlearning without imposing any a-priori asymmetries in the model: any asymmetric dynamicsthat arise in my model are due to the information structure. The assumption of private signalsturns out to have implications on these asymmetries and it also allows the studying of dispersionin posterior beliefs.

Adding these complexities to the model requires simplifying elsewhere in order to keep themodel tractable. For this reason I take on the simplifying assumption of allowing Followerto observe Loner but not vice versa, while the literature usually explores homogeneous agentsand symmetric equilibriums.3 The model should thus be viewed as a step toward a completesolution of a problem that so far has not been addressed in the literature, i.e., social learning

3In settings of reversible actions, symmetric equilibriums are generally not monotone in believes and specif-ically, cutoff strategies are not an equilibrium. This is because in intermediate believes agents balance betweenfree-riding on others agents’ information (which lowers the incentive to enter the market) and encouraging oth-ers to engage (which increases the incentives to enter) [Horner and Skrzypacz (2016)]. By exogenously settingthe information sets of Loner and Follower I can guarantee that the only equilibrium is with cutoff strategies,even when I add the additional complexities of private information and stochastic state. This makes the modeltractable and facilitates the analysis.

4

with reversible actions, private information and a stochastic state. Nonetheless, it may also berelevant for settings in which agents’ abilities to observe others are exogenous to their investmentdecisions. For example, when two firms that engage in the same market differ in size it isplausible that the small firm will be able to observe the big firm’s actions but not vice versa.All the more so if one of the firms is private and the other is public and thus the latter isobligated to report its actions to shareholders. It may even be the case that two public firmsare subjected to different disclosure requirements. In Israel, for example, new firms on the stockexchange are granted leniencies in mandatory disclosure to shareholders.

The paper also relates to the literature that models the feedback between information andmacro or financial activity. The common feature of these models is that optimism boostseconomic activity which in turn generates more public information. Thus, there is a larger flowof information and faster reaction to shocks in good times, when shocks tend to be adverse,than in bad times, when shocks tend to be positive. This mechanism creates asymmetriccycles - the market crashes quickly following a negative shock but recovery is gradual. InVeldkamp (2005) this mechanism is modeled in a credit market where investment generatesinformation regarding the profitability of the project. Fajgelbaum et al. (2017) highlight the roleof uncertainty in investors’ decision-making and show that the interaction between informationand investment leads to uncertainty traps with low activity and high uncertainty. My modelfeatures similar mechanisms, as information is more abundant in good times when activity ishigh. However, since my focus is on information channels I relax the simplifying assumptionsmade in these models such as public information, myopic agents or irreversible entry to market.An asymmetric informational cycle also arises in Zeira (1994, 1999) and in Van Nieuwerburghand Veldkamp (2006). In these models the structure of the production function is such thatchanges in unobserved fundamentals are identified more accurately when production is high.

The rest of the paper is organized as follows. Section 2 describes the general setting of themodel; Section 3 discusses theoretical results and simulations for a setting with a constant stateof the world; Section 4 generalizes these results to a stochastic state setting; Section 5 analyzesthe role of private information by comparing the benchmark model to a public informationsetting; Section 6 discusses the dynamics of disagreement in equilibrium; Section 7 concludes.

2 General Setting

There are two possible states of the world θ ∈ {G,B} and time is discrete t = 0, 1, 2... . Initially,I assume that the state is constant over time and later I relax this assumption by allowing θt tofollow a Markov Chain process. The structure of the model is such that symmetry is maintainedwherever possible so that any asymmetric dynamics that arise are due solely to the informationstructure.

There are two risk neutral agents in the economy. Let µit denote the probability that agent i

assigns to the state of the world being G at the beginning of period t. I assume that priors, {µi0},

are common knowledge. Both agents discount future earnings by the same factor δ ∈ (0, 1).There is one asset in the economy and, conditional on the state θ, its returns are drawn

independently across time and across agents. Specifically, in each period the investment pays 0with probability p ∈ (0, 1), and πθ with probability 1− p. This means that if an agent observesπθ he learns the state of the world in that period, and a payoff of 0 teaches him nothing aboutthe state (note that 1−p, the probability of payment πθ, is independent of θ). This distributionof payoffs is chosen to maintain the a priori symmetry in the model as it imposes no trend inbeliefs.4 If an agent decides to invest in the asset in period t, he incurs a cost c in that period

4“One-armed bandit” models for example usually feature a Poisson distribution with a rate that depends onthe state of the world. However, the difference between rates imposes a drift in beliefs over time which governsaffects behavior.

5

and his payoff is private information. If he chooses not to invest his payoff in that period is 0.5

Denote by x the expected payoff from active investment in a single period given state G,meaning x = (1− p)πG − c. I assume that x > 0. In order to maintain symmetry between thetwo states of the world, I further assume that the expected payoff given state B is −x, meaning(1 − p)πB − c = −x < 0. This specifically means that the investment is profitable in state Gbut not in state B.

3 Constant State of the World

Assume that the state of the world θ is constant over time but initially unknown to agents.

3.1 Loner

As aforesaid, Loner is an isolated agent and cannot observe the other agent’s actions. Note thatthe only relevant information for his decision making in period t is his belief µt. This belief isupdated according to Bayes’ rule after each period’s payoff πt is revealed (note that πt = 0 ifthe agent did not invest or if his investment yielded a payoff of 0):

µt+1(πt) =

µt if πt = 01 if πt = πG

0 if πt = πB

The problem facing the agent is thus a dynamic optimal choice problem. Given a beliefµ, let V N (µ) and V I(µ) denote the value gained by the agent if he chooses not to invest inthe current period and to invest, respectively. The Bellman equation for the dynamic optimalchoice problem is therefore:

V (µ) = max{V N (µ), V I(µ)}

where,

V N (µt) = δV (µt)

V I(µt) = Eµt

[πt + δV (µt+1)]− c =

Pr(πt = 0|µt) [0 + δV (µt)]+Pr(πt = πG|µt)[πG + δV (1)

]+Pr(πt = πB|µt)

[πB + δV (0)

]− c =

pδV (µt)+(1−p)µt

[πG + δV (1)

]+(1−p)(1−µt)π

B−c = (2µt−1)x+δ

[

pV (µt) + (1− p)µtx

1− δ

]

Note that if Loner chooses not to invest, he will not receive new information, and thus if itis optimal for him not to invest now he will never invest again. The Bellman equation can thusbe written as:

V (µ) = max

{

0 , (2µ− 1)x+ δ

[

pV (µ) + (1− p)µx

1− δ

]}

(1)

I will now characterize V and show that Loner’s optimal choice is to invest as long as hisbelief exceeds a certain threshold.

5This setting can be viewed as follows. There is an outside option that guarantees a fixed payoff r (which wenormalize to 0). For the most part the investment is similar to the outside option and also pays r. The truenature of the investment is revealed only on rare occasions, with probability 1 − p, but in these occasions thepayoff is either extremely high or extremely low, depending on the state of the world. These rare occasions arewhat determine if the investment is preferable to the outside option.

6

Figure 1: Timeline of Investment Decisions and Payments

Loner

Follower

t t+1

!"# Action

I/N

Payment!

$"#

!"%&#

(Bayesian

update)

!"'

Action

I/N

Obs.

Loner’s

action

Payment

$"'

Definition 1. A function f : [0, 1] → R is threshold-linear convex (TLC) with thresholdM if f is continuous, increasing and convex, and if there exists a threshold M such that f islinear on [M, 1].

Proposition 1.

1. There exists a unique value function V : [0, 1] → R that satisfies (1).

2. There exists a threshold belief ML = 1−δ2−δ(1+p) such that Loner invests if and only if

µ ≥ ML. Furthermore, V is TLC with threshold ML (in fact, it is quasilinear).

Proof. See Proposition 3 for the general case.

Note that the agent’s threshold ML is smaller than 0.5 which is the threshold of the myopicagent (an agent with a discount factor δ = 0). This is due to the value of information embeddedin active investment. At the threshold belief, the agent is willing to incur a cost of (1−2ML)x =δ(1−p)x2−δ(1+p) > 0 for the opportunity to learn about the state of the world.

Another important insight is that there are only three types of histories that will materializein equilibrium:

1. Continued investment: If Loner’s prior µL0 was higher than ML then as long as he has

not seen a payoff of πB his belief is either µL0 or 1 and he continues to invest.

2. Investment terminated: If µL0 ≥ ML but at some point the agent observed πB then he

updated his belief to 0 and stopped investing.

3. Never invested: Of course, if µL0 < ML the agent will never invest.

3.2 Follower

To understand the role of learning by other’s actions, I now introduce a second agent to themodel - Follower. Like Loner, Follower can invest in the asset described above. Conditional onthe state this asset pays agents independent payoffs. While I assumed Loner can learn only byactive investment, Follower also observes Loner’s actions. These actions provide informationregarding Loner’s private signals and the state of the world. Specifically, in each period Followerobserves Loner’s investment decision before deciding on his own action. Figure 1 illustrates thetimeline of investment decisions and payments for both agents.

As pointed out in Section 3.1, there are three types of histories of Loner’s actions thatFollower may observe in equilibrium. If Loner never invested then Follower is an “isolated”

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agent and his dynamic optimal choice problem is as described in Section 3.1. If Loner investedin the past but at some point stopped investing, then Follower concludes that Loner observedπB. He immediately updates his own belief to 0 and stops investing (without ever resuminginvestment).

The interesting case is a history in which Loner invested in all previous periods. Whenobserving Loner investing one more period, Follower becomes more optimistic and his belief isupdated according to µ 7→ µ

p+(1−p)µ . Otherwise, his belief is updated to 0. If Follower does notinvest himself then this is his only source of information. If he does invest, then with probabilityp he receives a payment of zero and is left to learn only by Loner’s actions (as if he did notinvest in the first place). With probability 1 − p Follower’s own signal reveals the state of theworld and the information from Loner’s action is redundant.

Formally, denote Follower’s value function given Loner invested so far and given his ownaction a by Ua, a ∈ {I,N}. Then Follower’s Bellman equation is given by:

U(µ) = max{UN (µ), U I(µ)} where (2)

UN (µ) = [p+ (1− p)µ]δU

(µ

p+ (1− p)µ

)

U I(µ) = (2µ− 1)x+ pUN (µ) + (1− p)µδx

1− δ

Proposition 2.

1. There exists a unique value function U : [0, 1] → R that satisfies (2). This function isnon-decreasing, continuous and convex.

2. U(µ) ≥ V (µ) for all µ ∈ [0, 1], meaning that Follower has positive gains from observingLoner’s actions.

3. There exists a threshold belief MF = 1−δp2−δp(1+p) ∈ (ML, 0.5) such that Follower invests if

and only if µ ≥ MF , and U(·) is TLC with threshold MF . Specifically,

U(µ) = max

{

[p+(1−p)µ]δU

(µ

p+ (1− p)µ

)

,

[x

1− δ+

x

1− δp2

]

µ−x

1− δp2

}

(3)

Proof. See Proposition 4 for the general case.

The fact that MF > ML is due to Follower’s “free-riding”. At the belief ML an isolatedagent would have started investing but since Follower has the privilege of observing Loner, thegain from not investing is higher which makes this option more attractive. However, MF isstill lower than the threshold of the myopic agent, 0.5, since investing always provides excessinformation regarding the state of the world.

3.3 Dynamics

The simple setting of constant state provides insights that will carry through to the stochasticsettings. As I will demonstrate, Follower’s “free-riding” induces asymmetric dynamics in beliefsand investment. While the ascent of beliefs and investment occurs gradually, their declinecan be abrupt. Furthermore, agents’ optimism is accompanied by dispersion in beliefs, whilepessimistic beliefs are aligned.

An example of the dynamics arising in this model is depicted in Figure 2 (general resultsare established in Section 4.2). In this example both agents have a prior belief that is aboveLoner’s threshold but below Follower’s. Thus, Loner starts investing in period 0. In the firstfive periods Loner does not receive any information, meaning he does not observe πθ, his belief

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Figure 2: The Constant-State Model - Numeric Approximation of Value Functions and Simu-lation of Beliefs and Total Investment when Loner Receives Bad News

Notes: Panel A depicts numeric approximations of Loner’s and Follower’s value functions for the parameters

x = 1, δ = 0.95 and p = 0.8. Panels B and C depict the dynamics in a simulation of this model in which both

agents have a prior belief of 0.25 ∈ (ML,MF ) and no news arrive until period 4 (both agents receive payments

of zero). In period 5 Loner observes πB and leaves the market in the following period. His exit is immediately

followed by Follower’s departure from the market.

remains unchanged and he continues to invest. As for Follower, initially his belief is not highenough to evoke investment and he prefers to passively observe Loner (“free-riding” effect).Observing Loner staying in the market over time makes Follower gradually more optimisticuntil his belief crosses his own threshold and he enters the market. In Follower’s point of view,“no news is good news”, meaning that observing Loner staying in the market makes him moreconfident that the state of the world is G.

In the example of Figure 2 both agents receive payments of zero until period 4, meaningthat no new information arrives to the market. Nonetheless, Follower becomes highly optimisticand his and Loner’s beliefs grow further and further apart. In other words, the disagreementbetween the agents intensifies. In period 5 Loner observes πB which results in him updatinghis belief to 0 in period 6 and exiting the market. At this point Follower experiences a “WileE. Coyote moment”. Like the cartoon character who runs off the cliff in pursuit of the RoadRunner only to find himself hovering in mid-air, Follower suddenly realizes that his optimisticbelief is groundless and he immediately exits the market. From this point on, both agents sharethe same pessimistic belief, i.e., disagreement vanishes, and investment is halted indefinitely.

4 Stochastic State

In this section I explore a more general setting in which the state of the world varies overtime. Specifically, I assume that θt is a stationary Markov Chain process with the state space{G,B} and conditional probabilities Pr(θt = θ|θt−1 = θ) = λ for θ ∈ {G,B} and for someλ ∈ (0.5, 1]. This means that the state of the world is persistent (in each period there is ahigher probability to remain in the same state than to switch) but transition to the other stateis equally likely under both states. With no other information, beliefs evolve according toµ 7→ λµ+ (1− λ)(1− µ) = (2λ− 1)µ+ 1− λ and any initial belief converges over time to thesteady state 0.5.

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4.1 Loner and Follower’s Value Functions

The generalization of Loner’s value function to the stochastic state setting is straightforward. Itonly requires adjusting the evolution of beliefs to the possibility of state transition. Specifically,Loner’s value function takes the following form:

V (µ) = max{V N (µ), V I(µ)} where, (4)

V N (µ) = δV(

(2λ− 1)µ+ 1− λ)

V I(µ) = (2µ− 1)x+ pV N (µ) + δ(1− p)[µV (λ) + (1− µ)V (1− λ)]

The following proposition characterizes the function V and shows that there exists a uniquethreshold belief ML ∈ (1 − λ, 0.5) such that Loner invests only when his belief exceeds thisthreshold.

Proposition 3. (Properties of Loner’s Value Function)

1. There exist unique value functions V : [0, 1] → R+ that satisfies (4). This function isnon-decreasing, continuous and convex.

2. There exists a threshold belief ML ∈ (1 − λ, 0.5) such that Loner invests if and only ifµ ≥ ML, and his value function is TLC with threshold ML. Specifically,

V (µ) = max

{

δV(

(2λ− 1)µ+ 1− λ)

, αLµ+ βL

}

where αL = x(2−δ−δp)−(1−δ)(1−p)V (0)(1−δλ)(1−δp)+δp(1−δ)(1−λ) , β

L = x−(1−δλ)αL

1−δ, and ML = x−(1−p)[V (0)−βL]

(1+p)x−(1−p)[V (0)−βL]

Proof. All proofs are deferred to the Appendix.

Note that Loner’s belief evolves depending on periodic payoffs as follows:

µLt+1(πt) =

(2λ− 1)µLt + 1− λ if πt = 0λ if πt = πG

1− λ if πt = πB

Further note that if µLt ≥ ML then (2λ − 1)µL

t + 1 − λ ≥ ML as well.6 Therefore, Lonerwill continue to invest as long he has not received bad news, i.e., has not observed πB. Once heobserves πB his belief is updated in the following period to 1− λ and he stops investing. WhileµLt < ML Loner does not invest and his belief evolves according to µt+1 = (2λ− 1)µt + 1− λ.

This means that n periods after Loner observed πB his belief is 0.5 − 0.5(2λ − 1)n. Since thisprocess converges to 0.5, eventually Loner’s belief crosses his threshold and he starts investing.

More precisely, after observing πB Loner will not invest for n ≡ ⌊ ln(1−2ML)ln(2λ−1) ⌋ periods and will

resume investment in the (n+ 1)th period.This means that in equilibrium, in each period t there are two types of states that are

relevant to Follower:

1. Loner will resume investment in n periods (n ∈ {1, ..., n}): This means that Lonerdid not invest for n+ 1− n periods up until and including period t. In this case Loner’saction in t+ 1 is known for certain (if n = 1 he will invest in the following period, and ifn > 1 he will not) so it will be uninformative and Follower will update his belief accordingto µF

t+1 = (2λ− 1)µFt + 1− λ.

6If µ ≥ 0.5 then (2λ− 1)µ+ 1− λ ≥ 0.5, and if ML ≤ µ < 0.5 then (2λ− 1)µ+ 1− λ > µ.

10

2. Loner invested in t: In this case Follower’s belief µFt+1 is distributed according to

the probability that Loner will invest another period or will stop investing (see proof ofProposition 4 for a formal definition).

Formally, the possible states that Follower may be facing can be categorized by a couple(µF

t , nt) where µFt is Follower’s belief, and nt ∈ {0, 1, ..., n} is the number of periods remaining

until Loner resumes investment (nt = 0 indicates that Loner invested in t).Follower updates his belief µF

t according to his private signals and Loner’s actions as in theconstant state setting, with the additional adjustment to the possibility of state transition. Theprominent difference from the constant state setting is in the evolution of nt. While Loner isinvesting (nt = 0), Follower’s expected utility depends on whether Loner will remain in themarket for another period. If he does, then the state variable nt+1 remains zero. Otherwise,Follower’s deduces that Loner observed πB in period t and nt+1 updates to n. If nt ≥ 1then necessarily nt+1 = nt − 1 (the “counter” for the number of periods until Loner resumesinvestment decreases by 1). Thus, Follower’s value function has a recursive structure:

U(µ, n) = max{

UN (µ, n), U I(µ, n)}

where, (5)

UN (µ, 0) =[p+ (1− p)µ]δU

(

(2λ− 1)µ

p+ (1− p)µ+ 1− λ, 0

)

+ (1− p)(1− µ)δU(1− λ, n),

U I(µ, 0) =(2µ− 1)x+ pUN (µ, 0)+

(1− p)µδU(λ, 0) + (1− p)(1− µ)δ[

pU(1− λ, 0) + (1− p)U(1− λ, n)]

and for n ∈ {1, ..., n},

UN (µ, n) =δU(

(2λ− 1)µ+ 1− λ, n− 1)

,

U I(µ, n) =(2µ− 1)x+ pUN (µ, n) + (1− p)δ[

µU(λ, n− 1

)+ (1− µ)U

(1− λ, n− 1

)]

An important difference between the stochastic state setting and the constant state settingis that in the latter if an agent stopped investing he will never resume investment. However,when the state is stochastic even a belief of zero will eventually converge to 0.5 in which it isclearly optimal for the agent to invest. Inter alia, this implies that V (0) and U(0, n) are nolonger equal to zero.

The following proposition characterizes Follower’s value function and shows that for everyn there exists a unique threshold belief MF

n ∈ (1− λ, 0.5) such that Follower invests only whenhis belief exceeds this threshold.

Proposition 4. (Properties of Follower’s Value Function)

1. There exists a unique value function U : [0, 1] × {0, ..., n} → R+ that satisfies (5). Thisfunction is non-decreasing, continuous and convex in the first argument.

2. U(µ, n) ≥ V (µ) for all n = 0, ..., n and µ ∈ [0, 1].

3. For all n = 0, ..., n there exists a unique threshold belief MFn ∈ (1 − λ, 0.5) such that

Follower invests in state (µ, n) if and only if µ ≥ MFn , and U(µ, n) is TLC .

4. Define U(µ, n) as in (2) for all n ∈ N (i.e, extend the definition for n+ 1, n+ 2, ...) thenlimn→∞

U(µ, n) = V (µ). This means that as Loner’s entry is further in the future, Follower

acts as an isolated agent.

11

Figure 3 depicts numeric approximations of Loner and Follower’s thresholds as a functionof n - the number of periods until Loner is known to enter the market in equilibrium. Thefigure illustrates that when Loner’s entry is close (n is small) Follower’s incentive to invest islower than Loner’s and his threshold exceeds ML. This is because free information will becomeavailable in the near future. As n increases this promise becomes less valuable and Follower’sincentive to actively invest increases (his threshold decreases). Note that for some values of nFollower’s threshold is lower than ML. This is because the promise of Loner’s entry makes itless costly for Follower to enter the market and receive a bad outcome. To be more concrete,consider a belief of 0.3 when n = 12. Note that 0.3 ∈ (MF

12,ML). An isolated agent like Loner

that is facing a belief of 0.3 prefers to stay out of the market since his entry will likely resultin a low signal that will keep him out of the market for a long time (22 periods). However,Follower knows that in any case Loner will enter the market in 12 periods. When this occurs,Follower will receive information that will potentially be positive. Thus, Follower’s expectedloss from entering today and receiving bad news is lower. When n is large the promise offuture information has a very small effect on Follower’s incentives and his threshold for entryapproaches that of an isolated agent ML.

Even though Follower’s threshold may be lower than Loner’s for some values of n, Section4.2 will show that for λ close enough to 1 Follower will enter the market after Loner.

Figure 3: Numeric Approximation of Agents’ Investment Thresholds

Notes: The figure depicts numeric approximations of Loner and Follower’s thresholds (ML and MF

n ,

respectively) for the parameters x = 1, δ = 0.95, p = 0.8 and λ = 0.98. Thresholds are depicted as a function of

n, the number of periods until Loner is known to return to the market.

4.2 Asymmetric Dynamics

In this section I discuss the dynamics that arise in equilibrium. The pattern of investment inequilibrium tends to be asymmetric. When the state of the world is persistent enough, entry tothe market is gradual, meaning that Loner always enters before Follower. However, exit fromthe market is generally abrupt, meaning that both agents leave the market at the same time.Furthermore, after both agents leave recovery takes a long time, meaning that investment isre-initiated long after it ceased.

First, I will establish that entry to the market is gradual when λ is close enough to 1.7 This

7The focus on large λ’s is for two reasons. First, unlike the case of λ = 1, for λ < 1 thresholds cannot be

12

is due to Follower’s “free-riding”, meaning that in equilibrium he will prefer to wait and seeLoner stay in the market for a while before entering himself:

Proposition 5. (Free-Riding and Gradual Entry)Assume both agents share the same belief and Loner is out of the market. If λ is close

enough to 1 then Follower will enter the market strictly after Loner.

Next, I show that exit from the market is more likely to be abrupt than gradual, meaningthat the probability that both agents exit in the same period in equilibrium is higher than theprobability that they exist one after the other. Note that the former event occurs wheneverLoner observes a bad signal and exits the market, since Follower will immediately update hisbelief to 1− λ and exit as well. The latter event occurs only if Follower observes bad news butLoner observes an uninformative signal.

Proposition 6. (Abrupt vs. Gradual Exit)Assume both agents were in the market in period t− 1 and some agent leaves the market in

period t. The probability of an abrupt exit in t (both agents leave the market) is 11+p

while the

probability that only one agent exits is smaller and equals p1+p

.

Proof. In equilibrium, an exit occurs in t with probability (1 − p2)1{θt=B} where 1{θt=B} isthe indicator function that equals one if θt = B and zero otherwise. Given an exit at t, the

probability that Loner exits and thus the exit is abrupt, equals(1−p)1{θt=B}

(1−p2)1{θt=B}= 1

1+p.

Finally, after both agents leave the market recovery will be slow. Specifically, it will take nperiods for Loner to re-enter the market and if λ is high, Follower will enter even longer afterthat.

Corollary 7. (Slow Recovery)

If λ is close enough to 1, no agent will enter the market in the n = ⌊ ln(1−2ML)ln(2λ−1) ⌋ periods after

Loner exited.

An example of the asymmetric dynamics in the model is depicted in Figure 4. In thisexample, both agents have a prior belief that is lower than Loner’s threshold so initially theyare both out of the market. As time progresses their belief updates according to the Markovianprocess until it crosses ML in period 4. At this point, Loner enters the market but Followerprefers to “wait and see” how events unfold. In this example, Loner does not receive any news inperiods 4-7 so he remains in the market. Observing Loner continuing to invest makes Followermore and more optimistic until his belief crosses his threshold and he enters the market in period6. Note that this dynamic of gradual entry occurs without any actual news being received inthe market. In period 8, Loner is assumed to observe an adverse signal, causing him to updatehis belief downwards and exit the market. This leads Follower to experience a “Wile E. Coyotemoment” and exit as well. Consequently, activity plunges. Note that in period 31, i.e., n = 22periods after observing πB, Loner resumes investment and a new cycle of investment and learninginitiates. As in the constant state setting, one can notice that disagreement between agents ismore prominent when activity is high and private information is generated. In contrast, whenactivity plunges due to Loner leaving the market, disagreement also vanishes.

derived analytically and the approximation to λ = 1 allows to characterize them. Second, as demonstrated inFigure 3, Followers thresholds can be lower than those of Loner making Follower willing to“test the ground”before Loner enters. This is not the case when λ is close to 1 since high persistence of the state makes “testingthe ground” more costly.

13

Figure 4: The Stochastic-State Model - Numeric Approximation of Value Functions and Simu-lation of Beliefs and Total Investment when Loner Receives Bad News

Notes: Panel A depicts numeric approximations of Loner and Follower’s value functions for the parameters

x = 1, δ = 0.95, p = 0.8 and λ = 0.98. For Follower, the approximation is of U(µ, 0), i.e., his value function

while Loner is investing. Panels B and C depict the dynamics in a simulation of this model in which both agents

have a prior belief of 0.27 < ML and no news arrive until Loner observes πB in period 8. In panel B solid lines

represent agents’ beliefs and dashed lines represent thresholds. Note that Follower’s threshold is state-dependent

(see Figure 3 and associated discussion).

4.3 When Follower Chooses to Test the Waters

Proposition 5 established that Follower enters the market after Loner if the state of the worldis persistent enough, i.e., when λ is close to one. In this section I demonstrate that if the stateis more transitory, it may be that Follower will choose to “test the waters” and enter beforeLoner in equilibrium.

Panels C and D in Figure 5 depict a simulation of the model with λ = 2/3, i.e., with a moretransitory state than in the previous examples (other parameters are the same: p = 0.1, x = 1and δ = 0.95). In this case Loner renters the market n = 2 periods after he exits. The priorbelief in the simulation is assumed to be such that Loner enters the market in period 0 andFollower stays out. Furthermore, it is assumed that Loner observes a bad signal in period 0but afterwards no meaningful signal arrives to the market. Following the arrival of bad news,Loner exits the market in period 1 leading to the update of both agents’ beliefs to 1− λ = 1/3in period 1. In period 2 both agents update their belief according to the Markovian processto 4/9 which is above Follower’s threshold but below Loner’s. At this point Follower enters themarket without observing any reassuring news from Loner. This is because Follower knows thatLoner will enter the market in the next period so the cost of “testing the waters” is relativelylow. Without observing good news in period 2, Follower is not optimistic enough to stay in themarket and he exits in period 3. He renters only after observing Loner staying in the marketfor two periods.

5 The Public Information Case

The private information setting of previous sections endogenously generates asymmetry in thequality of information embodied in Loner’s actions. While an exit decisively indicates that Lonerobserved a bad signal, staying in the market only indicates that Loner has not yet observedsuch a signal, meaning that so far he received payoffs of zero and πG. One might wonder if thedifference in information quality is not responsible for some (or all) of the asymmetric dynamics

14

Figure 5: Numeric Approximation of Value Functions and Simulation of Beliefs and Investmentwith a Transitory State of the World

Notes: Panel A depicts numeric approximations of Loner and Follower’s value functions for the parameters

x = 1, δ = 0.95, p = 0.1 and λ = 2/3. For Follower, the approximation is of U(µ, 0), i.e., his value function while

Loner is investing. Panel B depicts numeric approximations of agents’ thresholds as a function of the number of

periods remaining until Loner enters the market, n. Panels C and D depict the dynamics in a simulation of this

model in which both agents have a prior belief of 0.47 ∈ (ML,MF

0 ), Loner observes a bad signal in period 0 and

afterwards no news arrive to either agent. Panel C depicts the evolution of agents’ beliefs (solid lines) and

numeric approximations of their thresholds (dashed lines). Panel D depicts active investors in each period.

in the model.For this purpose in this section I compare Follower’s behavior to a model with public in-

formation (I still restrict attention to a setting in which Follower observes Loner but not viceversa). I assume that there are two followers in the market: Follower A that observes Loner’sactions as in the baseline model, and Follower B that observes Loner’s signals. By allowingFollower B to observe Loner’s signals I break the asymmetric quality of information that isgenerated with private information.

As it turns out, this has no qualitative effect on the main results. If λ is close enough to1, Follower B still enters the market strictly after Loner and exits with him if Loner observes abad signal. They key insight is that gradual entry is due solely to the free-riding effect which isof course also relevant for Follower B. An abrupt exit occurs when Follower realizes that Lonerobserved a bad signal and this happens whether he sees Loner receive that signal or just seeshim leave the market.

15

However, public information does affect Follower’s incentive to enter and the magnitudeof the asymmetry in dynamics in the since that it affects the timing of Follower’s entry tothe market (after Loner has already entered). As I will show in this section, since Follower Bobserves information of higher quality when Loner is in the market, his cost from entering andobserving adverse news is lower than Follower A’s and thus his threshold for entry is also lower.However, if the state is persistent enough this difference is small and does not stimulate moreinvestment in equilibrium. In fact, generally Follower B enters the market after Follower A,i.e., public information defers entry and experimentation. Since the result of abrupt exit is stillmaintained, this means that public information magnifies the asymmetry in dynamics.

5.1 Follower B

Denote Follower B’s value function by UB(µ, n). It differs from Follower A’s value function onlywhen Loner is in the market and Follower does not observe a meaningful signal (i.e, receives apayoff of zero). This is because when Loner is out of the market both followers are exposed tothe same information, and when their signal is informative the information they observe fromLoner is redundant. In other words, UB differs from U in UN

B (µ, 0).Formally, Follower B’s value function is defined as follows:

UB(µ, n) = max{UNB (µ, n), U I

B(µ, n)}

where,

UNB (µ, 0) =pδUB

(

(2λ− 1)µ+ 1− λ, 0)

+ (1− p)δ[

µUB(λ, 0) + (1− µ)UB(1− λ, n)]

,

U IB(µ, 0) =(2µ− 1)x+ pUN

B (µ, 0)+

(1− p)µδUB(λ, 0) + (1− p)(1− µ)δ[

pUB(1− λ, 0) + (1− p)UB(1− λ, n)]

and for n ∈ {1, ..., n},

UNB (µ, n) =δUB

(

(2λ− 1)µ+ 1− λ, n− 1)

,

U IB(µ, n) =(2µ− 1)x+ pUN

B (µ, n) + (1− p)δ[

µUB

(λ, n− 1

)+ (1− µ)UB

(1− λ, n− 1

)]

Similar arguments to those given in the proof of Proposition 4 can be used to show thatit holds for UB as well, meaning that Follower B’s value function shares the same propertiesas Follower A’s value function. However, because Follower B observes information of higherquality, his value function is higher than Follower A’s:

Proposition 8. UB(µ, n) ≥ U(µ, n) for all n = 0, ..., n and µ ∈ [0, 1].

5.2 The Effect of Public Information on Follower’s Incentives and on Asym-metric Dynamics

First, I will formally show that even though public information eliminates differences in signalquality that arise with private information, it still generates asymmetric dynamics. Note thatthe results of abrupt versus gradual entry (Proposition 6) and slow recovery (Corollary 7) stillhold in the public information setting, i.e., if Follower A is substituted by Follower B. Tocomplete the argument of asymmetric dynamics, the following proposition shows that FollowerB also free-rides on Loner’s information and enters the market after him when the state of theworld is persistent enough:

16

Figure 6: Public vs. Private information - Simulation of Beliefs and Total Investment whenLoner Receives Bad News

Notes: The figure depicts a simulation of the model with parameters x = 1, δ = 0.95, p = 0.8 and λ = 0.98. The

simulation includes three agents: Loner, Follower A which observes Loner’s actions and Follower B which

observes Loner’s signals. In the simulation all agents have a prior belief of 0.27 < ML and no news arrive until

Loner observes πB in period 8. Panel B depicts the evolution of beliefs (solid lines) and thresholds (dotted

lines) which are state-dependent. Panel C depicts aggregate investment.

Proposition 9. (Free-Riding and Gradual Entry with Public Information)Assume Loner and Follower B share the same belief and Loner is out of the market. If λ is

close enough to 1 then Follower B will enter the market strictly after Loner.

Next, Proposition 10 highlights the differences between the two information settings: itcompares the two followers’ incentives to enter the market and their behavior in equilibrium.

Proposition 10. (Incentives and Entry - Public vs. Private Information)

1. Denote Follower B’s threshold when Loner is in the market (i.e., n = 0) by MFB0 , then

MFB0 ≤ MF

0 .

2. MFB0 = MF

0 when λ = 1.

3. In equilibrium, if λ is close enough to 1, Follower B enters the market before Follower Aif and only if Loner receives a good signal πG while both followers are out.

As aforesaid, the free-riding effect holds in both information settings and in equilibrium bothfollowers enter the market after Loner. The question then remains which follower enters firstafter Loner? The first item in Proposition 10 states that while Loner is in the market Follower Bhas a (weakly) lower threshold for investment compared to Follower A. This is because FollowerB has a smaller cost for entering the market and receiving bad news. Nonetheless, item 3 impliesthat Follower B will generally enter after Follower A:

Corollary 11. Public information is more likely to defer entry and experimentation.In equilibrium, if λ is close enough to 1, the probability that Follower B will enter before FollowerA is smaller than 0.5.

Figure 6 depicts a similar simulation to the one from Figure 4, but this time with the twotypes of followers. Note that Follower B’s threshold is lower than A’s but it is quite close.

17

Further note that while Follower B is out of the market, his belief coincides with Loner’s.Because Loner doesn’t receive any news until period 8 in this example, their beliefs remainrelatively low and do not cross Follower B’s threshold. Thus, Follower B does not even enterthe market. However, Follower A is more optimistic and his belief crosses his threshold aftertwo periods. This example demonstrates the general case in which public information defersentry and experimentation. Follower B would have entered before A only if Loner would havereceived a good signal in period 4 or 5, while Follower A is still out. But this is highly unlikely:

Pr(

πLt = πG ∨ πL

t+1 = πG∣∣∣Loner entered in t, θt

)

=

{(1− λ)(1− p) if θt = B

1− p+ pλ(1− p) if θt = G

Since the state is more likely to be B than G when Loner enters, the total probability isvery low (the general proof is given the Appendix):

Pr(

πLt = πG ∨ πL

t+1 = πG∣∣∣Loner entered in t

)

<

0.5[

(1− λ)(1− p) + 1− p+ pλ(1− p)]

= 0.5[

0.004 + 0.396]

= 0.2

6 Evolution of Disagreement over the Business Cycle

As demonstrated in the example of Figure 4, agents’ beliefs can vary substantially. Further-more, this variation is not constant over the business cycle. Specifically, agents’ beliefs becomeperfectly aligned once Loner exits the market: no matter if Follower is in or out at that point,Loner’s exit signals that he observed πB so both agents update their beliefs to 1− λ and theyremain aligned for the next n periods until Loner re-enters the market.

Before elaborating on the dynamics of disagreement, I should formally define what it is.Disagreement is a concept that stands for the dispersion of beliefs. It is therefore naturallycaptured by the variance in periodic beliefs V ar(µi

t) = 0.25(µLt −µF

t )2. To discuss the cyclicality

of this variable I compare it to total investment which is the number of agents in the marketand may equal 0, 1 or 2.

The above discussion implies that when total investment is zero then disagreement alsoequals zero. Of course, if some agent is in the market then he produces private signals whichcan drive beliefs apart, meaning that disagreement is positive when total investment is posi-tive. It follows that disagreement is expected to be pro-cyclical, i.e., positively correlated withinvestment.

Panel A in Table 2 shows this correlation in model simulations with different values of pand λ. It is apparent that for most parameter values this correlation is indeed positive. Theexceptions are low values of λ combined with high values of p which generate slightly negativecorrelations. For these values payoffs are extremely uninformative: when λ approaches 0.5 thestate is approximately white noise and when p approaches 1 payoffs are almost always 0 andare essentially independent of the state. Thus, in these settings disagreement is always verylow and slightly rises only when Follower leaves the market before Loner (i.e., total investmentequals 1), causing the small negative correlation.

To understand the role of the information structure in generating these results, Panel Bshows the correlation between investment and disagreement in simulations of a similar modelwith two isolated agents that do not observe each other’s actions. Comparing the two tablesshows that without the additional learning mechanism of observing another agent’s actions,the correlation between activity and disagreement falls and might even be very negative. Thisis true even though the second model also incorporates private signals which are generatedonly when activity is high. The lower correlation stems from periods of zero activity since any

18

Table 2: Correlation between Activity and Disagreement from Simulations with Different ModelParameters

A. Loner and Follower (Baseline)

λ

p

B. Two Loners

λ

p

Color Legend:

Note: Each entry in a table shows the correlation between total investment and disagreement V ar(µi

t) = 0.25(µF

t −µL

t )2 for different values of p and λ (note that λ is assumed throughout the paper to be larger than 0.5 and since

for λ = 1 a simulation futile I simulate over values of λ in (0.5, 1)). Panel A shows results from simulations ofthe baseline model with Loner and Follower; Panel B shows results from a model with two isolated agents thatdo not learn from each other. In all simulations: δ = 0.95, x = 1, number of observations = 106.

disagreement that was formulated when one agent left the market before the other is maintainedas long as they are both out. As aforesaid, this is not the case in the baseline model in whichzero activity necessarily means that beliefs are aligned.

7 Conclusion

The phenomenon of slow booms and sudden crashes is prevalent in many economic indicatorsof activity and beliefs (e.g., expectations). This paper showed that such asymmetries arise ina setting in which actions generate private information and agents learn from each other. Themechanism was studied for a simple setting of two agents: an isolated agent (Loner) and anobserver (Follower). In an otherwise completely symmetric setting, equilibrium exhibits asym-metric dynamics, namely, agents enter the market gradually but tend to exit simultaneously.Since Follower “free-rides” on the information generated by Loner, he prefers to observe himstay in the market for a while before entering himself. This is the cause for the gradual in-crease in activity. On the other hand, Follower acts immediately upon observing Loner exit themarket, as this action decisively indicates that Loner observed bad news.

19

In essence, this mechanism is relevant even when Follower observes Loner’s private signalsand not just his actions. However, the former case differs from the private information bench-mark in two important aspects. First, with public information Follower has access to superiorinformation which makes entry to the market less costly. Thus, his threshold for entry is lowerrelative to the benchmark model. Second, in the public information setting, Follower can tell ifLoner is in the market because he observed good news or because he had not yet observed badnews. Thus, Follower can wait to see actual good news before entering the market. Usually thesecond effect prevails and Follower delays his entry relative to the benchmark model.

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20

Appendix A Proofs

A.1 Proof of Proposition 3 (Properties of Loner’s Value Function)

1. Define the mappings:

TL(f)(µ) = max

{

δf(

(2λ− 1)µ+ 1− λ)

,

(2µ− 1)x+ pδf(

(2λ− 1)µ+ 1− λ)

+ δ(1− p)[µV (λ) + (1− µ)V (1− λ)]

}

It is easy to verify that TL satisfies Blackwell’s conditions for a contraction mappingand preserves monotonicity, continuity and convexity. It follows that this mapping has aunique fixed point, V , that is non-decreasing, continuous and convex.8

2. Since V ≥ 0 it is clear that Loner invests when µ ≥ 0.5. Thus there exists a thresholdML ≤ 0.5 such that V N (ML) = V I(ML) and Loner invests in [ML, 1]. Furthermore, notethat if µ ∈ [ML, 1] then also (2λ−1)µ+1−λ ∈ [ML, 1] (if µ ≥ 0.5 then (2λ−1)µ+1−λ ≥0.5, and if ML ≤ µ ≤ 0.5 then µ < (2λ− 1)µ+ 1− λ). This implies that ∀µ ∈ [ML, 1],

V I(µ) = (2µ− 1)x+ pδV I(

(2λ− 1)µ+ 1− λ)

+ δ(1− p)[

µV (λ) + (1− µ)V (1− λ)]

It can now be verified that in this segment V (µ) = αLµ+ βL for αL and βL as specified

in the proposition, and that ML = x−(1−p)[V (0)−βL](1+p)x−(1−p)[V (0)−βL]

.

Note that it must be that ML > 1−λ. Otherwise, if ML ≤ 1−λ, then V (µ) = αLµ+βL

for all µ ≥ 1− λ which implies that

V (µ) = max

{

δ[

αF(

(2λ− 1)µ+ 1− λ)

+ βF]

, (2µ−1)x+δ[

αF(

(2λ− 1)µ+ 1− λ)

+ βF]}

and thus ML = 0.5 but this is a contradiction since 0.5 > 1− λ.

It remains to show that ML is the unique threshold for investment. Let IGL(µ) =V I(µ) − V N (µ) denote Loner’s gain from investment. I will show that IGL is increasingwhich will imply that ML is the unique belief that nullifies IGL.

IGL(µ) = (2µ− 1)x+ (1− p)δ[

µV (λ) + (1− µ)V (1− λ)− V(

(2λ− 1)µ+ 1− λ)]

=

[(1 + p)µ− 1]x+ (1− p)[

µV (1) + (1− µ)V (0)− δV(

(2λ− 1)µ+ 1− λ)]

8Blackwell’s sufficient condition for a contraction: Let X ⊆ Rn and let B be a space of bounded functions

f : X → R. Let T : B → B be a mapping that satisfies: (1) Monotonicity: if f ≤ g then Tf ≤ Tg. (2)Discounting: there exists δ ∈ (0, 1) such that T [(f + a)](x) ≤ Tf(x) + δa for all f ∈ B, a ≥ 0, x ∈ X. Then T isa contraction mapping.If T satisfies these conditions and B is a complete space then following Banach’s fixed point theorem, T has aunique fixed point V in B. Furthermore, lim

n→∞Tn(f) = V for every f ∈ B.

In the context of Proposition 3, B is the space of bounded, non-decreasing, continuous and convex functions.Thus V ’s existence and its properties follow from the fact that TL is a contraction mapping and TL(B) ⊆ B.

21

IGL(µ+∆)− IGL(µ) = (1 + p)x∆+ (1− p)[

V (1)− V (0)]

∆−

(1− p)δ[

V(

(2λ− 1)(µ+∆) + 1− λ)

− V(

(2λ− 1)µ+ 1− λ)]

≥

(1 + p)x∆+ (1− p)[

V (1)− V (0)]

∆− (1− p)δαL(2λ− 1)∆ =

(1 + p)x∆+ (1− p)[

αL + βL − V (0)]

∆− (1− p)δαL(2λ− 1)∆ =

(1 + p)x∆+ (1− p)αL[

1− δ(2λ− 1)]

∆− (1− p)[

V (0)− βL]

∆

where the inequality in the second line is due to V ′s convexity which implies that the

maximal slope of V is αL. Now, since ML = x−(1−p)[V (0)−βL](1+p)x−(1−p)[V (0)−βL]

∈ (0, 0.5] it follows that

(1− p)[V (0)− βL] ≤ x, so:

[IGL(µ+∆)− IGL(µ)

]/∆ ≥ (1− p)αL

[

1− δ(2λ− 1)]

> 0

A.2 Proof of Proposition 4 (Properties of Follower’s Value Function)

Before turning to the proof of Proposition 4, I will prove two lemmas. Lemma 12 shows thatU(µ, 0), Follower’s value function when Loner is in the market, can be written as the sum of a“self contained” value function W and a linear function. By “self contained” I mean that Wis only a function of µ, as opposed to U(µ, 0) that also depends on U(µ, n). In other words, aFollower that maximizes according to W does not take into account that Loner may exit themarket. The purpose of this lemma is to facilitate the proofs of some of U(µ, 0)’s properties -those that are preserved when adding a linear function such as convexity and the threshold forinvestment. Lemma 13 is used to prove W ’s convexity.

Lemma 12. U(µ, 0) can be written as the sum of a linear function and a function W as follows:

U(µ, 0) = W (µ) + aµ+ b (6)

where,

W (µ) = max{WN (µ),W I(µ)},

WN (µ) = [p+ (1− p)µ]δW

(

(2λ− 1)µ

p+ (1− p)µ+ 1− λ

)

+ (1− p)(1− µ)δW (1− λ),

W I(µ) = (2µ− 1)x+ pWN (µ) + δ(1− p)[

µW (λ) + (1− µ)W (1− λ)]

and

[a, b] =(1− p)[W (0)− U(0, n)]

(1− δλ)(1− δp) + δp(1− δ)(1− λ)

[

1− δ, −(1− δλ)]

Proof. Denote f(µ) = U(µ, 0)−W (µ), thus:

U(µ, 0) = W (µ)+f(µ) = max

{

UN (µ)+

[

WN (µ)−UN (µ, 0)+f(µ)

]

, U I(µ, 0)+

[

W I(µ)−U I(µ)+f(µ)

]}

So necessarily f(µ) = UN (µ, 0) −WN (µ) = U I(µ, 0) −W I(µ). To find f I take an “educatedguess” that it is linear and solve the two equations:

aµ+ b = UN (µ, 0)−WN (µ) =

[p+(1−p)µ]δ

[

a(2λ− 1)µ

p+ (1− p)µ+ a(1− λ) + b

]

+(1−p)(1−µ)δ[U(1−λ, n)−W (1−λ)

]=

δ[

a(2λ− 1)µ+ [p+ (1− p)µ][a(1− λ) + b]]

+ (1− p)(1− µ)δ[U(1− λ, n)−W (1− λ)

]=

22

aµ+ b = U I(µ, 0)−W I(µ) =

p[aµ+b]+δ(1−p)[

µ(aλ+b)+(1−µ)[

pU(1−λ, n)−W (1−λ)]]

+(1−p)2(1−µ)δU(1−λ, n) =

p[aµ+b]+δ(1−p)[

µ(aλ+b)+(1−µ)p[

a(1−λ)+b]]

+(1−p)2(1−µ)δ[U(1−λ, n)−W (1−λ)

]

Both equations are equivalent to:

aµ+ b = δ[

µ(aλ+ b) + (1− µ)p[

a(1− λ) + b]]

+ (1− p)(1− µ)[U(0, n)−W (0)

]

which yields the solution:

[ab

]

=

[1− δλ 1− δ

−δp(1− λ) 1− δp

]−1

·

[0

(1− p)[U(0, n)−W (0)]

]

=

(1− p)[U(0, n)−W (0)]

(1− δλ)(1− δp) + δp(1− δ)(1− λ)

[−(1− δ)1− δλ

]

Lemma 13. Let f : A → R+ be a real function such that 0 ∈ A ⊆ R. Define g : A → R asfollows:

g(x; f) = [p+ (1− p)x]f

(

(2λ− 1)x

p+ (1− p)x+ 1− λ

)

If f is increasing and convex (concave) then g is also increasing and convex (concave).

Proof. Let 0 ≤ x < y ≤ 1 and c ∈ (0, 1). From the convexity of f it follows that,

f

(

(2λ− 1)cx+ (1− c)y

p+ (1− p)[cx+ (1− c)y]+ 1− λ

)

≤

f

(

(2λ− 1)x

p+ (1− p)x+ 1− λ

)

+(1− c)p[f(y)− f(x)](2λ− 1)(y − x)

[p+ (1− p)x][p+ (1− p)[cx+ (1− c)y]]

and that,

f

(

(2λ− 1)cx+ (1− c)y

p+ (1− p)[cx+ (1− c)y]+ 1− λ

)

≤

f

(

(2λ− 1)y

p+ (1− p)y+ 1− λ

)

−cp[f(y)− f(x)](2λ− 1)(y − x)

[p+ (1− p)y][p+ (1− p)[cx+ (1− c)y]]

Now,

g(cx+(1−c)y; f) =[

p+(1−p)[cx+(1−c)y]]

f

(

(2λ− 1)cx+ (1− c)y

p+ (1− p)[cx+ (1− c)y]+ 1− λ

)

=

c[p+ (1− p)x]f

(

(2λ− 1)cx+ (1− c)y

p+ (1− p)[cx+ (1− c)y]+ 1− λ

)

+

(1− c)[p+ (1− p)y]f

(

(2λ− 1)cx+ (1− c)y

p+ (1− p)[cx+ (1− c)y]+ 1− λ

)

≤

c[p+ (1− p)x]f

(

(2λ− 1)x

p+ (1− p)x+ 1− λ

)

+ c(1− c)p[f(y)− f(x)](2λ− 1)(y − x)

p+ (1− p)[cx+ (1− c)y]+

(1− c)[p+(1−p)y]f

(

(2λ− 1)y

p+ (1− p)y+ 1− λ

)

− (1− c)cp[f(y)− f(x)](2λ− 1)(y − x)

p+ (1− p)[cx+ (1− c)y]=

cg(x; f) + (1− c)g(y; f)

23

I now turn to the proof of Proposition 4.

Proposition 4.

1. There exists a unique value function U : [0, 1] × {0, ..., n} → R+ that satisfies (5). Thisfunction is non-decreasing, continuous and convex in the first argument.

2. U(µ, n) ≥ V (µ) for all n = 0, ..., n and µ ∈ [0, 1].

3. For all n = 0, ..., n there exists a unique threshold belief MFn ∈ (1 − λ, 0.5) such that

Follower invests in state (µ, n) if and only if µ ≥ MFn , and U(µ, n) is TLC .

4. Define U(µ, n) as in (5) for all n ∈ N (i.e, extend the definition for n + 1, n + 2, ...) thenlimn→∞

U(µ, n) = V (µ). This means that as Loner’s entry is further away, Follower acts as

an isolated agent.

Proof.

1. The set of states Follower may be facing is S = [0, 1]× {0, ..., n}. Denote by Qa(s, s′) theprobability of transition from state s′ to state s, depending on Followers action a.

If Follower chooses not to invest (a = N):

QN(

s, (µ, 0))

=

p+ (1− p)µ if s =(

(2λ− 1) µp+(1−p)µ + 1− λ, 0

)

(1− p)(1− µ) if s = (1− λ, n)

0 otherwise

For n ∈ {1, ..., n}, QN(

s, (µ, n))

=

{

1 if s =(

(2λ− 1)µ+ 1− λ, n− 1)

0 otherwise

If Follower chooses to invest (a = I):

QI(

s, (µ, 0))

=

p[p+ (1− p)µ] if s =(

(2λ− 1) µp+(1−p)µ + 1− λ, 0

)

(1− p)µ if s = (λ, 0)

(1− p)p(1− µ) if s = (1− λ, 0)

(1− p)(1− µ) if s = (1− λ, n)

0 otherwise

For n ∈ {1, ..., n}, QI(

s, (µ, n))

=

p if s =(

(2λ− 1)µ+ 1− λ, n− 1)

(1− p)µ if s = (λ, n− 1)

(1− p)(1− µ) if s = (1− λ, n− 1)

0 otherwise

Follower’s dynamic choice problem (5) can then be written as follows:

U(µ, n) = max

{

δ∑

s∈S

U(s)QN(

s, (µ, n))

, (2µ− 1)x+ δ∑

s∈S

U(s)QI(

s, (µ, n))}

Now, define the mapping:

TF (f)(µ, n) = max

{

δ∑

s∈S

f(s)QN(

s, (µ, n))

, (2µ− 1)x+ δ∑

s∈S

f(s)QI(

s, (µ, n))}

24

It is easy to verify that TF satisfies Blackwell’s condition for a contraction mapping andpreserves monotonicity and continuity in µ. It follows that this mapping has a uniquefixed point, U(µ, n), which is non-decreasing and continuous in µ.

To show that U(µ, 0) is convex in µ it suffices to show that W (µ) from Lemma 12 isconvex. Following Lemma 13, W (µ) is a maximum of two convex functions and thusconvex. Now, from the recursive structure of U(µ, n) for n = 1, ..., n it is clear that thesefunctions are also convex in µ.

2. Following Equation (5), note that for all n = 1, ..., n we can write U(µ, n) = TLU(µ, n−1)where TL is the mapping defined in Proposition 3 whose fixed point is V . Further note

that the space B ≡{

f |f : [0, 1] × {0, ..., n} → R

}

is homeomorphic to the space B ≡{

f |f : [0, 1] → Rn+1

}

. Thus, TF that is defined on B has an equivalent transformation

TF that operates on B and its fixed point is U(µ) ≡(U(µ, 0), ..., U(µ, n)

)′:

TF (f)(µ) = TF

f(µ, 0)f(µ, 1)

...f(µ, n)

≡

TF (f)(µ, 0)TF (f)(µ, 1)

...TF (f)(µ, n)

=

max{

δ∑

s∈S f(s)QN(

s, (µ, 0))

, (2µ− 1)x+ δ∑

s∈S f(s)QI(

s, (µ, 0))}

TL(f)(µ, 0)...

TL(f)(µ, n− 1)

Since U is TF ’s fixed point, in order to show that it is greater than V ≡(V, ..., V

)′, it

suffices to show that if f(µ) = (f(µ, 0), ..., f(µ, n))′ ≥ V (µ) then TF (f)(µ) ≥ V (µ) aswell.9

Since TL is monotone then TL(f)(µ, n) ≥ TL(V )(µ) = V (µ) for all n = 0, ..., n− 1. As forthe first argument of TF (f)(µ), it is the maximum of two arguments:

δ∑

s∈S

f(s)QN(

s, (µ, 0))

=

[p+ (1− p)µ]δf

(

(2λ− 1)µ

p+ (1− p)µ+ 1− λ, 0

)

+ (1− p)(1− µ)δf(1− λ, n) ≥

[p+ (1− p)µ]δV

(

(2λ− 1)µ

p+ (1− p)µ+ 1− λ

)

+ (1− p)(1− µ)δV (1− λ) ≥︸︷︷︸

V convex

δV ((2λ− 1)µ+ 1− λ) = V N (µ)

and,

(2µ− 1)x+ δ∑

s∈S

f(s)QI(

s, (µ, 0))

=

(2µ− 1)x+ p

[

δ∑

s∈S

f(s)QN(

s, (µ, 0))]

+ (1− p)µδf(λ, 0)+

(1− p)(1− µ)δ[

pf(1− λ, 0) + (1− p)f(1− λ, n)]

≥

(2µ− 1)x+ pV N (µ) + (1− p)µδV (λ) + (1− p)(1− µ)δV (1− λ) = V I(µ)

9If f ≥ V implies that TF (f) ≥ V then Tn

F (f) ≥ V for all n and in the limit U = limn→∞

Tn

F (f) ≥ V .

25

Thus, the first argument of TF (f)(µ) is greater than max{V N (µ), V I(µ)} = V (µ).

3. I will prove the claim by induction on n.

For n = 0:

Following Lemma 12 it suffices to prove the claim for W (since U(µ, 0) = W (µ) + aµ+ b,the unique threshold of W will also be the unique threshold of U(µ, 0)). The existence of athreshold for W follows from similar arguments to those given in the proof of Proposition3, i.e., it can be shown that there exists MF

0 ∈ (1− λ, 0.5) such that W I(µ) ≥ WN (µ) forall µ ∈ [MF

0 , 1] and that W is linear in this segment. Specifically, it can be shown that

W (µ) =

max

{

[p+(1−p)µ]δW

(

(2λ− 1)µ

p+ (1− p)µ+ 1− λ

)

+(1−p)(1−µ)δW (1−λ) , αFµ+βF

}

where αF = x(2−δ−δp2)−(1−δ)(1−p2)W (0)(1−δλ)(1−δp2)+δp2(1−δ)(1−λ)

, βF = x−(1−δλ)αF

1−δandMF

0 = x−(1−p)[W (0)−βF ](1+p)x−(1−p)[W (0)−βF ]

.

It remains to show thatMF0 is the unique threshold. Denote IGF (µ, 0) = W I(µ)−WN (µ).

I will show that IGF is increasing in µ which will imply that MF0 is the unique belief that

nullifies it. Since W is convex, αF is the maximal slope of any arch of W . Thus,

[WN (µ+∆)−WN (µ)]/δ =

[p+ (1− p)(µ+∆)]

W

(

(2λ− 1)µ+∆

p+ (1− p)(µ+∆)+ 1− λ

)

−

[

p+ (1− p)µ]

W

(

(2λ− 1)µ

p+ (1− p)µ+ 1− λ

)

− (1− p)∆W (1− λ) =

(1− p)∆

[

W

(

(2λ− 1)µ+∆

p+ (1− p)(µ+∆)+ 1− λ

)

−W (1− λ)

]

+

[

p+(1−p)µ] [

W

(

(2λ− 1)µ+∆

p+ (1− p)(µ+∆)+ 1− λ

)

−W

(

(2λ− 1)µ

p+ (1− p)µ+ 1− λ

)]

≤

(1− p)∆αF (2λ− 1)µ+∆

p+ (1− p)(µ+∆)+αF (2λ− 1)

p∆

p+ (1− p)(µ+∆)= αF (2λ− 1)∆

Now,

IGF (µ+∆, 0)−IGF (µ, 0) = (1+p)x∆+(1−p)[

W (1)−W (0)]

∆−(1−p)[

WN (µ+∆)−WN (µ)]

≥

(1 + p)x∆+ (1− p)[

αF + βF −W (0)]

∆− (1− p)δαF (2λ− 1)∆ ≥

(1− p)αF∆− (1− p)δαF (2λ− 1)∆ = (1− p)αF∆[

1− δ(2λ− 1)]

≥ 0

where the penultimate inequality is due to fact that MF0 < 0.5 so W (0)− βF ≤ (1 + p)x.

For n = 1, ..., n:

I will prove that U(µ, n) has a unique threshold for investment MFn ∈ (1−λ, 0.5) and that

26

U(µ, n) is TLC . Recall that:

U(µ, n) = max{

UN (µ, n), U I(µ, n)}

=

max

{

δU(

(2λ− 1)µ+ 1− λ, n− 1)

,

(2µ−1)x+pδU(

(2λ−1)µ+1−λ, n−1)

+δ(1−p)[µU(λ, n−1)+(1−µ)U(1−λ, n−1)]

}

Since U(µ, n− 1) is convex in µ then for µ ≥ 0.5,

(1−p)δU(

(2λ−1)µ+1−λ, n−1)

≤ (2µ−1)x+δ(1−p)[

µU(λ, n−1)+(1−µ)U(1−λ, n−1)]

This means that there exists MFn < 0.5 such that Follower invests in (µ, n) for µ ≥ MF

n .The uniqueness of MF

n follows from the fact that U(µ, n − 1) is convex, using similararguments to those given in the proof of the uniqueness of ML (Proposition 3).

It remains to show that U(µ, n) is TLC. For µ ≥ MFn ,

U(µ, n) = (2µ−1)x+pδU(

(2λ−1)µ+1−λ, n−1)

+δ(1−p)[

µU(λ, n−1)+(1−µ)U(1−λ, n−1)]

Since U(µ, n − 1) is TLC with some threshold M then U(µ, n) is TLC with threshold

max{

MFn , M−(1−λ)

2λ−1

}

. In fact, by induction we get that U(µ, n) is TLC with threshold

max{

MFn ,

MFn−1−(1−λ)

2λ−1 , ..., 0.5−0.5−MF

0(2λ−1)n

}

.

4. As aforesaid, for all n = 1, ..., n we can write

U(µ, n) = TLU(µ, n− 1) = (TL)nU(µ, 0) (7)

where TL is the mapping whose fixed point is V . This means that if we extend definition(7) to all n ∈ N (holding U(0, n) fixed) then lim

n→∞U(µ, n) = lim

n→∞(TL)

nU(µ, 0) = V (µ).10

A.3 Proof of Proposition 5 (Free-Riding and Gradual Entry)

Unfortunately, in the stochastic state setting agents’ thresholds cannot be derived analytically.The main difficulty is that an agent’s value in µ = 0 is not derived analytically since it dependson the number of periods until the agent resumes investment which in turn depends on thevalue at µ = 0. In the constant state model this was not an issue since the agent never returnedto the market after leaving so the value at µ = 0 was zero. Therefore, the proof of free-ridingin the stochastic case is based on an approximation of thresholds to the static state thresholds.Specifically, Lemma 14 shows that agents’ thresholds are continuous in λ at λ = 1. Thus wecan use the analytical solutions of agents’ thresholds at λ = 1 to show that the free-riding effectarises at the proximity of λ = 1. Lemmas 15 and 16 set bounds on the thresholds to facilitatethe proof.

Lemma 14. Think of Loner’s threshold an Follower’s threshold when Loner is in the marketas functions of λ and denote them by ML(λ) and MF

0 (λ), respectively. These thresholds arecontinuous in λ at λ = 1.

10Banach’s fixed point theorem guarantees that Tn

L (f) → V for any f . See Footnote 8.

27

Proof. Loner’s threshold: Consider the value of V at a belief µ as a function of λ as welland denote it by V (µ;λ). V (µ;λ) is convex in µ and bounded by the present value of a constantpayment x. It follows that

V (0;λ) = δV (1− λ;λ) ≤ (1− λ)δV (1;λ) + λδW (0;λ) ≤ δ(1− λ)x

1− δ+ λδV (0;λ)

Thus, 0 ≤ V (0;λ) ≤ δ(1−λ)x(1−δλ)(1−δ) for all λ and equality holds for λ = 1. It follows that V (0;λ)

is continuous in λ at (0; 1) and its value is zero at this point. Consequently, the followingexpression is also continuous at λ = 1:

DL(λ) ≡ V (0;λ)− βL = V (0;λ)−x− (1− δλ)αL

1− δ=

V (0;λ)

[

1−(1− δλ)(1− p)

(1− δλ)(1− δp) + δp(1− δ)(1− λ)

]

+1− δλ− δp(1− λ)

(1− δλ)(1− δp) + δp(1− δ)(1− λ)x (8)

Finally, it follows that ML(λ) = x−(1−p)DL(λ)(1+p)x−(1−p)DL(λ)

is continuous at λ = 1.

Follower’s threshold: Using similar arguments it can be shown that W (0;λ) is continuous

at λ = 1 which implies that DF (λ) ≡ W (0, λ) − βF and MF0 (λ) = x−(1−p)DF (λ)

(1+p)x−(1−p)DF (λ)are also

continuous at that point.

Lemma 15. Denote Dn = U(0, n)− βn and yn = x−(1−p)Dn

(1+p)x−(1−p)Dn for all n = 0, ..., n+ 1.

1. y0 = MF0 .

2. yn is the belief that solves αnyn + βn = δ(αn−1[(2λ− 1)yn + 1− λ] + βn−1

)and it can

be interpreted as follows. If Follower knew that if he would observe a signal 0 in all nconsecutive periods he will be investing in all of them (and thus will be on the linear part ofU in all the following n periods), then yn is the belief that makes him indifferent betweeninvesting and not investing today.

3. MFn ≥ yn .

Proof. 1. follows trivially from the solutions of MF . 2. follows from the recursive structure ofαn and βn:11

(αn

βn

)

= δ

(λ− p(1− λ) 1− pp(1− λ) p

)(αn−1

βn−1

)

+

(2x− (1− p)U(0, n)−x+ (1− p)U(0, n)

)

(9)

11Since U(µ, n) = UI(µ, n) is linear on [0.5, 1] for all n, the recursive structure can be achieved by solving

αnµ+βn = UI(µ, n) = (2µ−1)x+pδ(

αn−1[(2λ−1)µ+1−λ]+βn−1)

+(1−p)δ[

µ(αn+βn)+(1−µ)U(1−λ, n−1)]

for µ = 0.5, 1.

28

To prove 3. I need to show that U I(yn, n) ≤ UN (yn, n):

U I(yn, n) = (2yn − 1)x+ pUN (yn, n) + (1− p)δ[

ynU(λ, n− 1) + (1− yn)U(1− λ, n− 1)]

=

(2yn − 1)x+ pδ[

αn−1(

(2λ− 1)yn + 1− λ)

+ βn−1]

+

(1− p)δ[

ynU(λ, n− 1) + (1− yn)U(1− λ, n− 1)]

+

pUN (yn, n)− pδ[

αn−1(

(2λ− 1)yn + 1− λ)

+ βn−1]

=︸︷︷︸

Eq. (9)

αnyn + βn+

pUN (yn, n)− pδ[

αn−1(

(2λ− 1)yn + 1− λ)

+ βn−1]

=︸︷︷︸

Lemma 15.2

δ[

αn−1(

(2λ− 1)yn + 1− λ)

+ βn−1]

+

pUN (yn, n)− pδ[

αn−1(

(2λ− 1)yn + 1− λ)

+ βn−1]

=

pUN (yn, n) + (1− p)δ[

αn−1(

(2λ− 1)yn + 1− λ)

+ βn−1]

≤

pUN (yn, n) + (1− p)δU(

(2λ− 1)yn + 1− λ, n− 1)

= UN (yn, n)

Lemma 16. ML(λ) ≤ (1−δλ)(1−δ)+δ(1−λ)(2−δ−p)(2−δ−δp)[1−δ(2λ−1)] ≡ ML(λ)

Proof. Following Equation (8) and due to the fact that V (0) ≥ 0, I can boundDL(λ) from below:

DL(λ) ≥ 1−δλ−δp(1−λ)(1−δλ)(1−δp)+δp(1−δ)(1−λ)x. Now,ML(λ) = x−(1−p)DL(λ)

(1+p)x−(1−p)DL(λ)≤ (1−δλ)(1−δ)+δ(1−λ)(2−δ−p)

(2−δ−δp)[1−δ(2λ−1)] .

I now turn to the proof of Proposition 5:

Proposition 5. (Free-Riding and Gradual Entry)

Assume both agents share the same belief and Loner is out of the market. If λ is closeenough to 1 then Follower will enter the market strictly after Loner.

Proof.Loner will start investing in exactly n periods if and only if his belief today is in the region[

0.5− 0.5−ML

(2λ−1)n , 0.5−0.5−ML

(2λ−1)n−1

)

. Following Lemma 15, in order to show that Follower will not

enter before that, it suffices to show that zn−1 ≡ 0.5 − 0.5−ML

(2λ−1)n−1 ≤ yn for all n = 0, ..., n + 1.

The proof will be conducted by induction.

For n=0:For the purpose of this proof denote thresholds as functions of λ, i.e., ML(λ) and MF

0 (λ).I need to show that (2λ− 1)ML(λ) + (1− λ) ≤ MF (λ) for λ close enough to 1. The inequalityis strong for λ = 1 since ML(1) = 1−δ

2−δ(1+p) <1−δp

2−δp(1+p) = MF0 (1) (Propositions 1 and 2). Thus,

from the continuity of thresholds at λ = 1 (Lemma 14), it follows that the inequality holds inan environment of λ = 1.

For n+ 1 ≥ 1:Note that yn+1 = x−(1−p)Dn+1

(1+p)x−(1−p)Dn+1 ≥ zn if and only if Dn+1 ≤ 1−(1+p)zn

(1−p)(1−zn)x.

29

Dn+1 = x− pδ(1− λ)αn + pδ[U(1− λ, n)− βn

]=

x+ pδ[U(0, n)− βn

]− pδ(1− λ)

[

αn −U(1− λ, n)− U(0, n)

1− λ

]

≤︸︷︷︸

U is convex

x+ pδ[U(0, n)− βn

]− pδ(1− λ)

[

αn −αn0.5 + βn − U(0, n)

0.5

]

=

x+ pδ(2λ− 1)[U(0, n)− βn

]= x+ pδ(2λ− 1)Dn ≤

︸︷︷︸

ind. hyp.

x

[

1 + pδ(2λ− 1)1− (1 + p)zn−1

(1− p)(1− zn−1)

]

=︸︷︷︸

zn−1=(2λ−1)zn+1−λ

=

x

1 + pδ(2λ− 1)1− (1 + p)

[

(2λ− 1)zn + 1− λ]

(1− p)[

λ− (2λ− 1)zn]

It thus remains to show that:

1 + pδ(2λ− 1)1− (1 + p)

[

(2λ− 1)zn + 1− λ]

(1− p)[

λ− (2λ− 1)zn] ≤

1− (1 + p)zn

(1− p)(1− zn)

Rearranging this inequality I get that I need to prove:

∀n ∈ {1, ..., n+ 1} g(zn) ≡ 1− δ(2λ− 1) + pδ(2λ− 1)zn + 1−λ

2λ−1

1− zn + 1−λ2λ−1

−zn

1− zn≥ 0 (10)

I will prove that g(z) ≥ 0 for all z ∈ [0,ML]. Note that g′(z) = pδ

[ λ

2λ−1−z]

2 − 1[1−z]2

≤ 0 since

pδ ≤ 1 and λ2λ−1 ≥ 1 ≥ z. Thus, in order to prove (10) it suffices to show that g(ML) ≥ 0

where ML is the bound defined in Lemma 16.

g(z) ≥ 0 if and only if

[

1−δ(2λ−1)

] [

1− z +1− λ

2λ− 1

] [

1−z

]

+pδ(2λ−1)

[

z +1− λ

2λ− 1

] [

1−z

]

−z

[

1− z +1− λ

2λ− 1

]

≥ 0

Rearranging this inequality yields:

[

1− δ −(

2− δ(1 + p))

z]

︸ ︷︷ ︸

A(z)

+(1− λ)1− 2z

1− z

[1

2λ− 1+ δ(1 + p)(1− z)

]

︸ ︷︷ ︸

B(z)

≥ 0 (11)

Now,

A(ML) = 1− δ −(1− δλ)(1− δ) + δ(1− λ)(2− δ − p)

1− δ(2λ− 1)=

(1− δλ)(1− δ)− δ(1− λ)(2− δ − p)

1− δ(2λ− 1)

30

B(ML) = (1− λ)δ(1− δ)(1− p)(2λ− 1)

(1− δλ)(1− δp) + δp(1− δ)(1− λ)

[1

2λ− 1+ δ(1 + p)(1− ML)

]

≥

(1− λ)δ(1− δ)(1− p)(2λ− 1)

1− δλ+ (1− δ)(1− λ)

[1

2λ− 1+ 0

]

=δ(1− δ)(1− p)(1− λ)

1− δ(2λ− 1)

Thus,

A(ML) +B(ML) ≥(1− δλ)(1− δ)− δ(1− λ)(2− δ − p) + δ(1− δ)(1− p)(1− λ)

1− δ(2λ− 1)=

(1− δ)2 − δ2(1− p)(1− λ)

1− δ(2λ− 1)

It follows that inequality (11) is satisfied for ML if (1− p)(1− λ) ≤(1−δδ

)2.

A.4 Proofs on Private vs. Public Information

A.4.1 Proof of Proposition 8

Similarly to the proof of Proposition 4.2, in order to show that U(µ) =(U(µ, 0), ..., U(µ, n)

)′is

smaller than UB(µ) ≡(UB(µ, 0), ..., UB(µ, n)

)′, it suffices to show that if f(µ) = (f(µ, 0), ..., f(µ, n))′ ≤

UB(µ) then TF (f)(µ) ≤ UB(µ) as well (see proof of Proposition 4.2 for more details).Since TL is monotone then TF (f(µ))n = TL(f(µ, n− 1)) ≤ TL(UB(µ, n− 1)) = UB(µ, n) for

all n = 1, ..., n. As for the first argument of TF (f)(µ), it is the maximum of two arguments:

δ∑

s∈S

f(s)QN(

s, (µ, 0))

=

[p+ (1− p)µ]δf

(

(2λ− 1)µ

p+ (1− p)µ+ 1− λ, 0

)

+ (1− p)(1− µ)δf(1− λ, n) ≤

[p+ (1− p)µ]δUB

(

(2λ− 1)µ

p+ (1− p)µ+ 1− λ, 0

)

+ (1− p)(1− µ)δUB(1− λ, n) ≤︸︷︷︸

UB convex

pδUB

(

(2λ− 1)µ+ 1− λ, 0)

+ (1− p)δ[

µUB(λ, 0) + (1− µ)UB(1− λ, n)]

= UNB (µ, 0)

and,

(2µ− 1)x+ δ∑

s∈S

f(s)QI(

s, (µ, 0))

=

(2µ− 1)x+ p

[

δ∑

s∈S

f(s)QN(

s, (µ, 0))]

+ (1− p)µδf(λ, 0)+

(1− p)(1− µ)δ[

pf(1− λ, 0) + (1− p)f(1− λ, n)]

≤

(2µ− 1)x+ pUNB (µ, 0) + (1− p)µδUB(λ, 0)+

(1− p)(1− µ)δ[

pUB(1− λ, 0) + (1− p)UB(1− λ, n)]

= U IB(µ, 0)

Thus, the first argument of TF (f)(µ) is smaller than max{UNB (µ, 0), U I

B(µ, 0)} = UB(µ, 0).

31

A.4.2 Proof of Proposition 9

The proof requires three steps:

1. MFB0 (λ) is continuous at λ = 1

2. yBn ≡ x−(1−p)[UB(0,n)−βBn](1+p)x−(1−p)[UB(0,n)−βBn]

≤ MFBn

3. yBn ≥ zn−1 ≡ 0.5− 0.5−ML

(2λ−1)n−1 for λ close enough to 1.

The proofs are similar to the ones in the private information case (Appendix A.3) so I leaveout some details.

1. DefineWB(µ) = max{WN

B (µ),W IB(µ)}

where,

WNB (µ) = pδWB

(

(2λ− 1)µ+ 1− λ)

+ (1− p)δ[

µWB(λ) + (1− µ)WB(1− λ)]

W IB(µ) = (2µ− 1)x+ pWN

B (µ) + (1− p)δ[

µWB(λ) + (1− µ)WB(1− λ)]

Similarly to the proof of Proposition 4 (Appendix A.2) it can be shown that WB has the

same threshold as UB and it is equal to MFB0 = x−(1−p)[WB(0)−βFB ]

(1+p)x−(1−p)[WB(0)−βFB ]where βFB =

x−(1−δλ)αFB

1−δand αFB = x(2−δ−δp2)−(1−δ)(1−p2)WB(0)

(1−δλ)(1−δp2)+δp2(1−δ)(1−λ).

Similarly to the arguments given in Appendix A.3 it can be shown that if we think ofWB(0) as a function of λ then it is continuous at λ = 1, and so is MFB

0 .

2. Follows from the proof of Lemma 15 by replacing U with UB.

3. The proof is conducted by induction. For n = 0, due to the continuity of MF0 (λ) and

ML(λ) at λ = 1, it suffices to show that ML(1) = 1−δ2−δ(1+p) < MFB

0 (1). Note that if

λ = 1 then WB(0) = 0. By inserting this in the solution of MFB0 from item 1 we get that

MFB0 (1) = 1−δp

2−δp(1+p) which is in fact larger than ML(1). The proof for n ≥ 1 follows from

the proof of Proposition 5 (Appendix A.3) by replacing U, αn and βn with UB, αBn andβBn.

A.4.3 Proof of Proposition 10

1. The fact thatMFB0 ≤ MF

0 will follow from two claims: (1)WB ≥ W and (2)WB(0)−βB ≥W (0)− βF .

(1) Denote by TB the mapping whose fixed point is WB. It suffices to show that if f ≥ Wthen TB(f) ≥ W as well (see footnote 9). TB(f) is the maximum of two elements:

fN (µ) ≡ pδf(

(2λ− 1)µ+ 1− λ)

+ (1− p)δ[

µf(λ) + (1− µ)f(1− λ)]

≥

pδW(

(2λ− 1)µ+ 1− λ)

+ (1− p)δ[

µW (λ) + (1− µ)W (1− λ)]

≥︸︷︷︸

Wconvex

[p+(1−p)µ]δW

(

(2λ− 1)µ

p+ (1− p)µ+ 1− λ

)

+(1−p)(1−µ)δW (1−λ) = WN (µ)

32

f I(µ) ≡ (2µ− 1)x+ pfN (µ) + δ(1− p)[

µf(λ) + (1− µ)f(1− λ)]

≥

(2µ− 1)x+ pWN (µ) + δ(1− p)[

µW (λ) + (1− µ)W (1− λ)]

= W I(µ)

It follows that TB(f)(µ) = max{fN (µ), f I(µ)} ≥ max{WN (µ),W I(µ)} = W (µ)

(2) Using (1):

WB(0)− βB = WB(0)−x− (1− δλ)αB

1− δ=

WB(0)

[

1−(1− δλ)(1− δ)(1− p2)

(1− δλ)(1− δp2) + δp2(1− δ)(1− λ)

]

+

1− δλ− δp2(1− λ)

(1− δλ)(1− δp2) + δp2(1− δ)(1− λ)x ≥

W (0)

[

1−(1− δλ)(1− δ)(1− p2)

(1− δλ)(1− δp2) + δp2(1− δ)(1− λ)

]

+

1− δλ− δp2(1− λ)

(1− δλ)(1− δp2) + δp2(1− δ)(1− λ)x = W (0)− βF

2. If λ = 1 then claim (2) holds with equality since W (0) = WB(0) = 0. In this caseMFB

0 = MF0 = 1−δp

2−δp(1+p) .

3. Assume that Loner entered in some period that we normalize to 0. In equilibrium all agentsshare the same belief at this point, call it µ. Consider n periods after Loner’s entry, Loneris in the market and both types of followers were out at n− 1. Since Follower A observedLoner stay in the market for n periods, his belief in period n is χn ≡ µ

pn+(1−pn)µ . As

for Follower B, if Loner received a good signal πG in period n then Follower B updateshis belief to λ and enters, i.e., he enters (weakly) before Follower A. Otherwise, so farLoner received only uninformative signals of zero and Follower B has a belief of ζn(λ) ≡0.5− (2λ− 1)n(0.5− µ). I will show that

MF0 (λ)− χn ≤ MFB

0 (λ)− ζn(λ) (12)

This implies that if Follower A doesn’t enter in n (i.e., MF0 > χn), then Follower B also

doesn’t enter (i.e., MFB0 > ζn(λ)). Recall that MF

0 = MFB0 when λ = 1. Furthermore,

ζn(1) = µ < µpn+(1−pn)µ = χn. Thus, inequality (12) is strong when λ = 1 and from the

continuity of all elements (similarly to Lemma 14 it can be shown that MFB0 is continuous

at λ = 1), it holds in an environment of λ = 1.

A.4.4 Proof of Corollary 11

Following Proposition 10.4 I need to show that the probability that Loner observes πG beforeFollower A enters is smaller than 0.5. Assume Follower A enters k periods after Loner (in theleading example in the body of the paper k = 2). Let pθk denote the probability Loner observedπG in those k periods, without observing πB before that, given that the state was θ when heentered:

pθk = Pr(

πLt = ... = πL

t+i−1 = 0 , πLt+i = πGfor some i ≤ k

∣∣∣θt = θ, Loner entered in t

)

33

I need to show that∑

θ=G,B

pθk Pr(θt = θ|Loner entered in t) < 0.5 . Note that pGk and pBk

have the following recursive structure:

pGk =1− p+ p[

λpGk−1 + (1− λ)pBk−1

]

pBk =p[

λpBk−1 + (1− λ)pGk−1

]

pG0 =pB0 = 0

The solution for this system is:

pGk =2− pk(1 + p)− (2λ− 1)kpk(1− p)− 2λp(1− pk)

2[1− (2λ− 1)p

]

pBk =2p− pk(1 + p) + (2λ− 1)kpk(1− p)− 2λp(1− pk)

2[1− (2λ− 1)p

]

These expressions increase in k so pGk ≤ 1−p+(1−λ)p1−(2λ−1)p , pBk ≤ (1−λ)p

1−(2λ−1)p . Since in equilibriumLoner enters when the probability for state G is smaller than 0.5,

∑

θ=G,B

pθk Pr(θt = θ|Loner entered in t) < 0.5

1− p+ (1− λ)p

1− (2λ− 1)p+ 0.5

(1− λ)p

1− (2λ− 1)p= 0.5

34

��

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Published papers (in English)

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91.01��R. Melnick and Y. Golan - Measurement of Business Fluctuations in

Israel.��

91.03�� ���M. Sokoler - Seigniorage and Real Rates of Return in a Banking Economy.��

91.04E.K. Offenbacher - Tax Smoothing and Tests of Ricardian Equivalence

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91.07���M. Beenstock, Y. Lavi and S. Ribon - The Supply and Demand for

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91.08�� �R. Ablin - The Current Recession and Steps Required for Sustained

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91.10�� ���M. Beenstock - Business Sector Production in the Short and Long Run in

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91.13����A. Marom - The Black-Market Dollar Premium: The Case of Israel��

91.14�� �A. Bar-Ilan and A. Levy - Endogenous and Exoqenous Restrictions on

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91.15�The Market for Labor in Israel���M. Beentstock and S. Ribon��

92.02�� �O. Bar Efrat - Interest Rate Determination and Liberalization of

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92.03�� �Z. Sussman and D.Zakai - Wage Gaps between Senior and Junior

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92.05�� �O. L iviatan - The Impact of Real Shocks on Fiscal Redistribution and

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92.06�� �A. Bregman, M. Fuss and H. Regev - The Production and Cost Structure ��of the Israeli Industry: Evidence from Individual Firm Data.��

92.07�� ��M. Beenstock, Y. Lavi and A. Offenbacher - A Macroeconometric

Model for Israel 1962-1990: A Market Equilibrium Approach to

Aggregate Demand and Supply. ��

92.09�� �R. Melnick - Financial Services, Cointegration and the Demand for

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92.12���R. Melnick - Forecasting Short-Run Business Fluctuations in Israel��

92.13�� �K. Flug, N. Kasir and G. Ofer - The Absorption of Soviet Immigrants in to

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��

��

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93.01�� ���B. Eden - How to Subsidize Education and Achieve Voluntary Integration:

An Analysis of Voucher Systems. ��

93.07��A.Arnon, D. Gottlieb - An Economic Analysis of the Palestinian

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93.09�� �K. Flug, N. Kasir - The Absorption in the Labor Market of Immigrants

from the CIS - the Short Run.��

94.01��R. Ablin - Exchange Rate Systems, Incomes Policy and Stabilization Some

Short and Long-Run Considerations. ��

94.02��B.Eden - The Adjustment of Prices to Monetary Shocks When Trade is

Uncertain and Sequential. ��

94.04��K. Flug, Z. Hercowitz and A. Levi - A Small -Open-Economy Analysis of

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94.05��R. Melnick and E. Yashiv - The Macroeconomic Effects of Financial

Innovation: The Case of Israel.��

94.09����A. Blass - Are Israeli Stock Prices Too High��

94.10�� ���Arnon and J. Weinblatt - The Potential for Trade Between Israel the

Palestinians, and Jordan.��

94.13�� ���B. Eden - Inflation and Price Dispersion: An Analysis of Micro Data.��

94.16��B. Eden - Time Rigidities in The Adjustment of Prices to Monetary

Shocks: An Analysis of Micro Data.��

94.17O. Yosha - Privatizing Multi-Product Banks.��

����

95.01��B. Eden - Optimal Fiscal and Monetary Policy in a Baumol-Tobin Model.��

95.02�� ���B. Bar-Nathan, M. Beenstock and Y. Haitovsky - An Econometric Model

of The Israeli Housing Market.��

95.04��A. Arnon and A. Spivak - A Seigniorage Perspective on the Introduction

of a Palestinian Currency.��

95.06�� ���M. Bruno and R. Melnick - High Inflation Dynamics: Integrating Short-

Run Accommodation and Long-Run Steady-States.��

95.08��M. Strawczynski - Capital Accumulation in a Bequest Economy.��

95.11��A. Arnon and A. Spivak - Monetary Integration Between the Israeli,

Jordanian and Palestinian Economies.

� �

��

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��

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96.03��A. Blass and R. S. Grossman - A Harmful Guarantee? The 1983 Israel

Bank Shares Crisis Revisited.��

96.04�� ���Z. Sussman and D. Zakai - The Decentralization of Collective Bargaining

and Changes in the Compensation Structure in Israeli`s Public Sector.��

96.05���M. Strawczynski - Precautionary Savings and The Demand for Annuities��

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Israel: Empirical Analysis.��

96.10���M. Strawczynski - Social Insurance and The Optimum Piecewise Linear

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97.02M. Dahan and M. Strawczynski - The Optimal Non-Linear Income Tax.��

97.05H. Ber, Y. Yafeh and O. Yosha - Conflict of Interest in Universal Banking:

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97.08A. Blass, Y. Yafeh and O. Yosha - Corporate Governance in an Emerging

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98.01N. Liviatan and R. Melnick - Inflation and Disinflation by Steps in Israel.��

98.02A. Blass, Y. Yafeh - Vagabond Shoes Longing to Stray - Why Israeli

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98.04G. Bufman, L. Leiderman - Monetary Policy and Inflation in Israel.���

98.06Z. Hercowitz and M. Strawczynski - On the Cyclical Bias in Government

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99.04A. Brender - The Effect of Fiscal Performance on Local Government

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99.05S. Ribon and O. Yosha - Financial Liberalization and Competition In

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2000.01L. Leiderman and H. Bar-Or - Monetary Policy Rules and Transmission

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2000.03Z. Hercowitz and M. Strawczynski - Public-Debt / Output Guidelines:

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2000.04Y. Menashe and Y. Mealem - Measuring the Output Gap and its Influence

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2000.06��J. Djivre and S. Ribon - Inflation, Unemployment, the Exchange Rate and

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2000.09J. Djivre and S. Ribon - Monetary Policy, the Output Gap and Inflation:

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2000.10J. Braude - Age Structure and the Real Exchange Rate.��

2000.11Z. Sussman and D. Zakai - From Promoting the Worthy to

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2001.01H. Ber, A. Blass and O. Yosha - Monetary Transmission in an Open

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2001.03N. Liviatan - Tight Money and Central Bank Intervention

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2001.04Y. Lavi, N. Sussman - The Determination of Real Wages in the Long Run

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2001.06J. Braude - Generational Conflict? Some Cross-Country Evidence.

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2001.12��D. Romanov and N. Zussman - On the Dark Side of Welfare:

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2002.03A. Blass O. Peled and Y. Yafeh - The Determinants of Israel’s Cost of

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2002.04N. Liviatan - The Money-Injection Function of the Bank of Israel

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2002.07Y. Menashe - Investment under Productivity Uncertainty.��

2002.08T. Suchoy and A. Friedman - The NAIRU in Israel: an Unobserved

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2002.09R. Frish and N. Klein � Rules versus Discretion - A Disinflation Case.�

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2003.02S. Ribon - Is it labor, technology or monetary policy? The Israeli economy

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2003.05A. Marom, Y. Menashe and T. Suchoy - The State-of-The-Economy Index

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2003.07N. Liviatan and R. Frish Public Debt in a Long Term Discretionary

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2003.09D. Romanov - The Real Exchange Rate and the Balassa-Samuelson

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2003.10R. Frish - Restricting capital outflow - the political economy perspective. ��

2003.12��H. Ber and Y. Yafeh - Venture Capital Funds and Post-IPO Performance in

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2003.13N. Klein - Reputation and Indexation in an Inflation Targeting Framework.

2003.17��N. Liviatan - Fiscal Dominance and Monetary Dominance in the Israeli

Monetary Experience.��

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2004.04��A. Blass and S. Ribon - What Determines a Firm's Debt Composition –

An Empirical Investigation.

2004.06��R. Sharabany - Business Failures and Macroeconomic Risk Factors.

2004.07��N. Klein - Collective Bargaining and Its Effect on the Central Bank

Conservatism: Theory and Some Evidence. ��������

2004.10J. Braude and Y. Menashe - Does the Capital Intensity of Structural

Change Matter for Growth?�

2004.11S. Ribon - A� New Phillips Curve for Israel.

2004.13A. Barnea and J. Djivre - Changes in Monetary and Exchange Rate

Policies and the Transmission Mechanism in Israel, 1989.IV – 2002.I.�

2004.15M. Dahan and M. Strawczynski - The Optimal Asymptotic Income

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2004.16N. Klein - Long-Term Contracts as a Strategic Device.

2004.17D. Romanov - The corporation as a tax shelter: evidence from recent

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2004.18� ���

T. Suchoy - Does the Israeli Yield Spread Contain Leading Signals?

A Trial of Dynamic Ordered Probit. �

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2005.01��

A. Brender - Ethnic Segregation and the Quality of Local Government

in the Minorities Localities: Local Tax Collection in the Israeli-Arab

Municipalities as a Case Study.�

2005.02��

A. Zussman and N. Zussman - Targeted Killings: Evaluating the

Effectiveness of a Counterterrorism Policy. �

2005.03N. Liviatan and R. Frish - Interest on reserves and inflation. ��

2005.04��A. Brender and A. Drazen - Political Budget Cycles in New versus

Established Democracies.

2005.09Z. Hercowitz and M. Strawczynski - Government Spending Adjustment:

The OECD Since the 1990s.

2005.12Y. Menashe - Is the Firm-Level Relationship between Uncertainty

and Irreversible Investment Non- Linear?

2005.13A. Friedman - Wage Distribution and Economic Growth.

2006.01E. Barnea and N. Liviatan - Nominal Anchor: Economic and Statistical

Aspects.

2006.02��A. Zussman, N. Zussman and M. �rregaard Nielsen - Asset Market

Perspectives on the Israeli-Palestinian Conflict.

2006.04��N. Presman and A. Arnon - Commuting patterns in Israel 1991-2004.

2006.05G. Navon and I. Tojerow - The Effects of Rent-sharing On the Gender

Wage Gap in the Israeli Manufacturing Sector.

2006.11S. Abo Zaid - The Trade–Growth Relationship in Israel Revisited:

Evidence from Annual Data, 1960-2004.�

2007.04 M. Strawczynski and J. Zeira - Cyclicality of Fiscal Policy In Israel.�

2007.05 A. Friedman - A Cross-Country Comparison of the Minimum Wage.

2007.08 K� Flug and M. Strawczynski - Persistent Growth Episodes and

Macroeconomic Policy Performance in Israel.�

2008.04 S. Miaari, A. Zussman and N. Zussman – Ethnic Conflict and Job

Separations.�

2008.07 Y. Mazar – Testing Self-Selection in Transitions between the Public

Sector and the Business Sector/�

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2008.08 A. Brender and L. Gallo – The Response of Voluntary and Involuntary

Female Part-Time Workers to Changes in Labor-Market Conditions.�

2008.10 A. Friedman – Taxation and Wage Subsidies in a Search-Equilibrium

Labor Market.�

2008.11 N. Zussman and S. Tsur – The Effect of Students' Socioeconomic

Background on Their Achievements in Matriculation Examinations.

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2009.02��R. Frish and N. Zussman – The Causal Effect of Parents' Childhood

Environment and Education on Their Children's Education.

2009.04��

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E. Argov – The Choice of a Foreign Price Measure in a Bayesian

Estimated New-Keynesian Model for Israel.

2009.05 G. Navon – Human Capital Spillovers in the Workplace: Labor

Diversity and Productivity. �

2009.06 T. Suhoy – Query Indices and a 2008 Downturn: Israeli Data.

2009.07 J. Djivre and Y. Menashe – Testing for Constant Returns to Scale and

Perfect Competition in the Israeli Economy, 1980–2006.

2009.08 S. Ribon – Core Inflation Indices for Israel

2009.10 A. Brender – Distributive Effects of Israel's Pension System.

2009.12 R. Frish and G. Navon – The Effect of Israel’s Encouragement of Capital

Investments in Industry Law on Product, Employment, and Investment:

an Empirical Analysis of Micro Data.

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2009.13 E. Toledano, R. Frish, N. Zussman, and D. Gottlieb – The Effect of Child

Allowances on Fertility.

2010.01 D. Nathan – An Upgrade of the Models Used To Forecast ��

the Distribution of the Exchange Rate.

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2010.02 A. Sorezcky - Real Effects of Financial Signals and Surprises.�

2010.03 B. Z. Schreiber – Decomposition of the ILS/USD Exchange Rate into

Global and Local Components��

2010.04 E. Azoulay and S. Ribon – A Basic Structural VAR of Monetary Policy in

Israel Using Monthly Frequency Data.

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2010.05 S. Tsur and N. Zussman – The Contribution of Vocational High School

Studies to Educational Achievement and Success in the Labor Market.

2010.06 Y. Menashe and Y. Lavi – The Long- and Short-Term Factors Affecting

Investment in Israel's Business-Sector, 1968-2008.��

2010.07 Y. Mazar – The Effect of Fiscal Policy and its Components on GDP

in Israel. �

2010.08 P. Dovman – Business Cycles in Israel and Macroeconomic Crises—Their

Duration and Severity.

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2010.09 T. Suhoy – Monthly Assessments of Private Consumption. �

2010.10 A. Sorezcky – Did the Bank of Israel Influence the Exchange Rate?

2010.11��Y. Djivre and Y. Yakhin – A Constrained Dynamic Model for

Macroeconomic Projection in Israel.����

2010.12 Y.� Mazar and N. Michelson – The Wage Differentials between Men and

Women in Public Administration in Israel – An Analysis Based on Cross-

Sectional and Panel Data.

2010.13 A. Friedman and Z. Hercowitz – A Real Model of the Israeli Economy.��

2010.14 M. Graham – CEO Compensation at Publicly Traded Companies.��

2010.15 E. Azoulay and R. Shahrabani – The Level of Leverage in Quoted

Companies and Its Relation to Various Economic Factors.

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2010.16 G. Dafnai and J. Sidi – �Nowcasting Israel GDP Using High Frequency

Macroeconomic Disaggregates.���

2010.17 E. Toledano, N. Zussman, R. Frish and D. Gottleib – Family Income and

Birth Weight.��

2010.18 N. Blass, S. Tsur and N. Zussman – The Allocation of Teachers' Working

Hours in Primary Education, 2001–2009.

2011.01 T. Krief – A Nowcasting Model for GDP and its Components.

2011.02 W. Nagar – Persistent Undershooting of the Inflation Target During

Disinflation in Israel: Inflation Avoidance Preferences or a HiddenTarget?

2011.03 R. Stein – Estimating the Expected Natural Interest Rate Using Affine

Term-Structure Models: The Case of Israel.

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2011.04R. Sharabani and Y. Menashe – The Hotel Market in Israel.

2011.05A. Brender – First Year of the Mandatory Pension Arrangement:

Compliance with the Arrangement as an Indication of its Potential

Implications for Labor Supply.

2011.06P. Dovman, S. Ribon and Y. Yakhin – The Housing Market in Israel

2008-2010: Are Housing Prices a “Bubble”?

2011.07 N. Steinberg and Y. Porath – Chasing Their Tails: Inflow Momentum

and Yield Chasing among Provident Fund Investors in Israel.

2011.08 L. Gallo – Export and Productivity – Evidence From Israel.

2011.09 I. Gamrasni – The Effect of the 2006 Market Makers Reform on the

Liquidity of Local-Currency Unindexed Israeli Government Bonds in the

Secondary Market.

2011.11 H. Etkes – The Impact of Employment in Israel on the Palestinian

Labor Force�

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2011.12 S. Ribon – The Effect of Monetary Policy on Inflation: A Factor

Augmented VAR Approach using disaggregated data��

2011.13 A. Sasi-Brodesky – Assessing Default Risk of Israeli Companies Using a

Structural Model.

2012.01 Y. Mazar and O. Peled – The Minimum Wage, Wage Distribution and

the Gender Wage Gap in Israel 1990–2009��

2012.02 N. Michelson – The Effect of Personal Contracts in Public Administration in

Israel on Length of Service.

2012.04 S. Miaari, A. Zussman and N. Zussman – Employment Restrictions and

Political Violence in the Israeli-Palestinian Conflict.��

2012.05 E. Yashiv and N. K. (Kaliner) – Arab Women in the Israeli Labor Market:

Characteristics and Policy Proposals.

2012.06 E. Argov, E. Barnea, A. Binyamini, E. Borenstein, D. Elkayam and

I. Rozenshtrom – MOISE: A DSGE Model for the Israeli Economy��

2012.07 E. Argov, A. Binyamini� E. Borenstein and I. Rozenshtrom – Ex-Post

Evaluation of Monetary Policy.�

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2012.08 G. Yeshurun – A Long School Day and Mothers' Labor Supply.

2012.09��Y. Mazar and M. Haran – Fiscal Policy and the Current Account.

2012.10 E. Toledano and N. Zussman – Terror and Birth Weight.��

2012.11 A. Binyamini and T. Larom – Encouraging Participation in a Labor Market

with Search and Matching Frictions.��

2012.12 E. Shachar –The Effect of Childcare Cost on the Labor Supply of Mothers

with Young Children.

2012.13 G. Navon and D. Chernichovsky – Private Expenditure on Healthcare,�Income Distribution, and Poverty in Israel.

2012.14 Z. Naor – Heterogeneous Discount Factor, Education Subsidy, and

Inequality.

2012.15 H. Zalkinder – Measuring Stress and Risks to the Financial System in Israel

on a Radar Chart.

2013.01 Y. Saadon and M. Graham – A Composite Index for Tracking Financial

Markets in Israel.

2013.02 A. Binyamini – Labor Market Frictions and Optimal Monetary Policy.

2013.03 E. Borenstein and D. Elkayam – The equity premium in a small open

economy, and an application to Israel.

2013.04 D. Elkayam and A. Ilek – Estimating the NAIRU for Israel, 1992–2011.

2013.05 Y. Yakhin and N. Presman�– A Flow-Accounting Model of the Labor

Market: An Application to Israel.

2013.06M. Kahn and S. Ribon – The Effect of Home and Rent Prices on Private

Consumption in Israel—A Micro Data Analysis.

2013.07 S. Ribon and D. Sayag – Price Setting Behavior in Israel – An Empirical

Analysis Using Microdata.

2013.08 D. Orpaig (Flikier) – The Weighting of the Bank of Israel CPI

Forecast—a Unified Model.

2013.09 O. Sade, R. Stein and Z. Wiener – Israeli Treasury Auction Reform.

2013.10 D. Elkayam and A. Ilek– Estimating the NAIRU using both the Phillips

and the Beveridge curves.

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2014.01 A. Ilek and G. Segal – Optimal monetary policy under heterogeneous

beliefs of the central bank and the public.

2014.02 A. Brender and M. Strawczynski – Government Support for Young

Families in Israel.

2014.03 Y. Mazar – The Development of Wages in the Public Sector and

their Correlation with Wages�in the Private Sector.

2014.06��H. Etkes – Do Monthly Labor Force Surveys Affect Interviewees’ Labor

Market Behavior? Evidence from Israel’s Transition from Quarterly to

Monthly Surveys.��

2014.07��N. Blass, S. Tsur and N. Zussman – Segregation of students in primary

and middle schools.��

2014.08 A. Brender and E. Politzer – The Effect of Legislated Tax Changes

on Tax Revenues in Israel.

2015.01

D. Sayag and N. Zussman – The Distribution of Rental Assistance

Between Tenants and Landlords: The Case of Students in Central

Jerusalem.

2015.02 A. Brender and S. Ribon –The Effect of Fiscal and Monetary Policies and

the Global Economy on Real Yields of Israel Government Bonds.

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2015.05 I. Caspi – Testing for a Housing Bubble at the National and Regional

Level: The Case of Israel.

2015.06 R. Stein – The Effects of Taxation of Capital Gains on the Pricing of

Financial Assets.

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2015.07��E. Barnea and Y. Menashe – Banks Strategies and Credit Spreads as

Leading Indicators for the Real Business Cycles Fluctuations.

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2016.01 R. Frish – Currency Crises and Real Exchange Rate Depreciation�

2016.02V. Lavy and E. Sand – On the Origins of Gender Gaps in Human Capital:

Short and Long Term Consequences of Teachers’ Biases.

2016.03R. Frish – The Real Exchange Rate in the Long Term.

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2016.04 A. Mantzura and B. Schrieber –�Carry trade attractiveness: A time-varying

currency risk premium approach.��

2016.05N. Blass, S. Tsur, and N. Zussman – Local-Authority Funding of Primary

School Instruction hours and Its Effect on Affirmative Action in the State

(Jewish, Non-Religious) Education System.

2016.06I. Caspi and M Graham – Testing for Bubbles in Stock Markets

With Irregular Dividend Distribution.�

2016.09E. Borenstein, and D. Elkayam – Financial Distress and Unconventional

Monetary Policy in Financially Open Economies.

2016.10 S. Afek and N. Steinberg – The Foreign Exposure of Public Companies

Traded on the Tel Aviv Stock Exchange.

2016.12 R. Stein – Review of the Reference Rate in Israel�Telbor and Makam

Markets.�

��2016.13 N. Sussmana and O. Zoharb – Has Inflation Targeting Become Less

Credible� Oil Prices, Global Aggregate Demand and Inflation �

Expectations during the Global Financial Crisis.

2016.14E. Demalach, D. Genesove A. Zussman and N Zussman – The Effect of

Proximity to Cellular Sites on Housing Prices in Israel. ���

2016.15 E. Argov – The Development of Education in Israel and its

Contribution to Long-Term Growth Discussion Paper. �

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2017.01S. Tsur – Liquidity Constraints and Human Capital: The Impact of Welfare

Policy on Arab Families in Israel.

2017.02G. Segal – Interest Rate in the Objective Function of the Central

Bank and Monetary Policy Design.

2017.03N. Tzur-Ilan – The Effect of Credit Constraints on Housing Choices: The

Case of LTV limit.

2017.04A. Barak – The private consumption function in Israel

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2017.04S. Ribon – Why the Bank of Israel Intervenes in the Foreign Exchange

Market, and What Happens to the Exchange Rate.�

2017.05E. Chen, I. Gavious and N. Steinberg – Dividends from Unrealized

Earnings and Default Risk.

2017.06A. Ilek and I. Rozenshtrom – The Term Premium in a Small Open

Economy : A Micro-Founded Approach.

72017.0 A Sasi-Brodesky – Recovery Rates in the Israeli Corporate Bond Market

2008–2015.

2017.09D. Orfaig – A Structural VAR Model for Estimating the Link between

Monetary Policy and Home Prices in Israel.

102017.M. Haran Rosen and O. Sade – Does Financial Regulation Unintentionally

Ignore Less Privileged Populations? The Investigation of a Regulatory

Fintech Advancement, Objective and Subjective Financial Literacy.

112017.E. Demalach and N. Zussman – The Effect of Vocational Education on

Short- and Long-Term Outcomes of Students: Evidence from the Arab

Education System in Israel.

2017.14S. Igdalov, R. Frish and N. Zussman – The Wage Response to a Reduction

in Income Tax Rates: The 2003–2009 Tax Reform in Israel.

2018.01Y. Mazar – Differences in Skill Levels of Educated Workers between the

Public and Private Sectors, the Return to Skills and the Connection

between them: Evidence from the PIAAC Surveys.

2018.03M. Graham-Rozen and N. Michelson – To Accept or Not to Accept?

Considerations in Providing Credit Insurance.

2018.04Itamar Caspi, Amit Friedman, and Sigal Ribon – The Immediate Impact

and Persistent Effect of FX Purchases on the Exchange Rate.

2018.05D. Elkayam and G Segal – Estimated Natural Rate of Interest in an

Open Economy: The Case of Israel.

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2018.07Y. Mazar and Y. Reingewertz – The Effect of Child Allowances on Labor

Supply Evidence from Israel.

2018.08����������������������������� ���!�"���#$�%��������&�'�����–N. B. Itzhak�����(�� �#$�����

2018.10 M. Rosenboim, Y. Saadon, B. Z. Schreiberc – “Much Ado about Nothing”?

The Effect of Print Media Tone on Stock Indices.

2018.11K. Kosenko and N. Michelson – It Takes More than Two to Tango:

Understanding the Dynamics behind Multiple Bank Lending and its

Implications.

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2018.12N. Tzur-Ilan – LTV Limits and Borrower Risk.

2019.03A. Barak – The Effect of Public Transit on Employment in Israel's

Arab Society

2019.04E. Argov and S. Tsur – A Long-Run Growth Model for Israel.

2019.05E. Argov and S. Tsur –Conditional Convergence and Future TFP

Growth in Israel.

2019.06O. Zohar – Boom-Bust Cycles of Learning, Investment and

Disagreement.

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