Flexible information acquisition and optimal Tobin tax in tractable dynamic global games

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FUNDAÇÃO GETULIO VARGAS ESCOLA DE ECONOMIA DE SÃO PAULO

RODRIGO DOS SANTOS BARBOSA

FLEXIBLE INFORMATION ACQUISITION AND OPTIMAL TOBIN TAX IN TRACTABLE DYNAMIC GLOBAL GAMES

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FUNDAÇÃO GETULIO VARGAS ESCOLA DE ECONOMIA DE SÃO PAULO

Tese submetida à Escola de Economia de São Paulo da Fundação Getulio Vargas, como requisito para obtenção do título de Doutor em Economia.

Orientador: Prof. Dr. Braz Ministério de Camargo

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Barbosa, Rodrigo dos Santos.

Flexible information acquisition and optimal Tobin tax in tractable dynamic global games / Rodrigo dos Santos Barbosa. - 2016.

125 f.

Orientadores: Braz Ministério de Camargo, Bernardo de Vasconcellos Guimarães

Tese (doutorado) - Escola de Economia de São Paulo.

1. Câmbio a termo - Impostos. 2. Finanças internacionais. 3. Liquidez (Economia). I. Camargo, Braz Ministério de. II. Guimarães, Bernardo de Vasconcellos. III. Tese (doutorado) - Escola de Economia de São Paulo. IV. Título.

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Tese submetida à Escola de Economia de São Paulo da Fundação Getulio Vargas, como requisito para obtenção do título de Doutor em Economia.

Data de aprovação: Banca examinadora:

——————————————

Prof. Dr. Braz Ministério de Camargo (Co-orientador) FGV-EESP

——————————————

Prof. Dr. Bernardo de Vasconcellos Guimara˜es

(Co-orientador) FGV-EESP —————————————— Prof. Dr. Daniel Monte

FGV-EESP

——————————————

Prof. Dr. Marcos Yamada Nakaguma FEA-USP

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AGRADECIMENTOS

Ao professor Bernardo Guimarães por tornar esta tese possível com sua orientação, apoio e paciência. Também a EESP-FGV e seus professores pelo excelente aprendizado que me proporcionaram durante todo este processo.

Aos amigos Flavio de Stéfani, Lucas Teixeira, Luis Fernando Azevedo, Marcos Ross e Renan Pieri pelo apoio e discussões a respeito deste trabalho.

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ABSTRACT

My dissertation focuses on dynamic aspects of coordination processes such as reversibility of early actions, option to delay decisions, and learning of the environment from the observation of other people’s actions. This study proposes the use of tractable dynamic global games where players privately and passively learn about their actions’ true payoﬀs and are able to adjust early investment decisions to the arrival of new information to investigate the consequences of the presence of liquidity shocks to the performance of a Tobin tax as a policy intended to foster coordination success (chapter 1), and the adequacy of the use of a Tobin tax in order to reduce an economy’s vulnerability to sudden stops (chapter 2). Then, it analyzes players’ incentive to acquire costly information in a sequential decision setting (chapter 3).

In chapter 1, a continuum of foreign agents decide whether to enter or not in an investment project. A fractionλof them are hit by liquidity restrictions in a second period and are forced to withdraw early investment or precluded from investing in the interim period, depending on the actions they chose in the first period. Players not affected by the liquidity shock are able to revise early decisions. Coordination success is increasing in the aggregate investment and decreasing in the aggregate volume of capital exit. Without liquidity shocks, aggregate investment is (in a pivotal contingency) invariant to frictions like a tax on short term capitals. In this case, a Tobin tax always increases success incidence. In the presence of liquidity shocks, this invariance result no longer holds in equilibrium. A Tobin tax becomes harmful to aggregate investment, which may reduces success incidence if the economy does not benefit enough from avoiding capital reversals. It is shown that the Tobin tax that maximizes the ex-ante probability of successfully coordinated investment is decreasing in the liquidity shock.

Chapter 2 studies the effects of a Tobin tax in the same setting of the global game model proposed in chapter 1, with the exception that the liquidity shock is considered stochastic, i.e, there is also aggregate uncertainty about the extension of the liquidity restrictions. It identifies conditions under which, in the unique equilibrium of the model with low probability of liquidity shocks but large dry-ups, a Tobin tax is welfare improving, helping agents to coordinate on the good outcome. The model provides a rationale for a Tobin tax on economies that are prone to sudden stops. The optimal Tobin tax tends to be larger when capital reversals are more harmful and when the fraction of agents hit by liquidity shocks is smaller. Chapter 3 focuses on information acquisition in a sequential decision game with payoff complementar-ity and information externalcomplementar-ity. When information is cheap relatively to players’ incentive to coordinate actions, only the first player chooses to process information; the second player learns about the true payoff distribution from the observation of the first player’s decision and follows her action. Miscoordination requires that both players privately precess information, which tends to happen when it is expensive and the prior knowledge about the distribution of the payoffs has a large variance.

Keywords: Coordination; Capital controls; Tobin tax; Liquidity shocks; Information acquisition;

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RESUMO

A presente tese concentra-se em aspectos dinâmicos de processos que envolvem coordenação entre agentes em ambientes com interação estratégica. Propomos utilizar os chamadosglobal games para estudar a

capacidade de uma Tobin tax elevar a probabilidade de sucesso em um ambiente em que investidores

internacionais sujeitos a choques de liquidez precisam coordenar suas decisões de investimento (capítulo 1), e reduzir a vulnerabilidade de uma economia aberta a ﬂuxos internacionais de capitais asudden stops

(capítulo 2). Também, investigamos o problema da aquisição de informação em jogos sequenciais com informação incompleta e complementaridade em ações (capítulo 3).

No capítulo 1, agentes estrangeiros decidem se entram ou não em um projeto, cujo sucesso depende em parte da capacidade dos mesmos em coordenarem suas escolhas. Uma fraçãoλdesses investidores é afetada por restrições de liquidez no segundo período do modelo e é forçada a se retirar do projeto ou impedida de entrar, dependendo de suas respectivas escolhas no primeiro período. Agentes não afetados pelo choque de liquidez possuem a opção de reavaliar decisões tomadas no primeiro estágio do jogo. É assumido que a probabilidade de sucesso do projeto de investimento é crescente no volume total de capital que a economia recebe, mas decrescente no volume de capitais que deixa a economia no segundo período. Na ausência de choques de liquidez (λ= 0), o volume de capital que é recebido em um estado pivotal para o sucesso do projeto de investimento independe da existência de um imposto sobre capitais de curto prazo. Como tal imposto sempre desestimula saídas de capitais, umaTobin tax sempre favorece as chances de

sucesso em uma economia em queλ= 0. Contudo, na presença de choques de liquidez, o volume total de investimento que a economia recebe torna-se decrescente em um imposto incidente sobre capitais de curto prazo. Neste caso, umaTobin tax pode prejudicar as chances do processo de coordenação ser bem

sucedido, caso o benefício de reduzir o volume de saída de capitais não seja suﬁcientemente grande. O capítulo 2 estuda os efeitos de uma Tobin tax no mesmo cenário do capítulo 1, porém considera

que a extensão da restrição de liquidez a que os agentes podem estar sujeitos é aleatória. Neste modelo, identiﬁcamos condições sob as quais uma Tobin tax reduz a probabilidade de se observar um sudden stope eleva o bem estar no único equilíbrio de uma economia onde a probabilidade de ocorrência de um

choque de liquidez é pequena, mas a magnitude de tal choque pode ser signiﬁcativa.

O capítulo ﬁnal investiga o problema de aquisição de informação em um jogo sequencial com 2 agentes, externalidade informacional e complementaridade em ações. Demonstramos que, quando o custo de aquisição de informação é pequeno relativamente ao incentivo que os agentes possuem para coordenarem suas ações, apenas o primeiro jogador escolhe adquirir novas informações a respeito da distribuição dos

payoffs, e o jogador 2 sempre segue a ação escolhida pelo jogador 1. Probabilidade positiva de se observar

divergência em ações requer que ambos os jogadores processem informação privadamente, o que tende a ocorrer quando o custo de aquisição de informação é baixo e a distribuiição a priori dos payoffspossui

variância elevada.

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List of Tables

3.1 Investment Return . . . 78

List of Figures

1.1 Capital ﬂows withλ= 0 . . . 18 1.2 Capital ﬂows withλ >0 . . . 19

2.1 Game’s tree . . . 39 2.2 Threshold fundamental in the good state and success probability withB1= 3,B2= 2.41,

γ=.085,λ=.5, andp=.1. . . . 51 2.3 Threshold fundamental in the good state and success probability withB1= 3,B2= 2.41,

γ=.085,λ=.5, andp=.05. . . . 51 2.4 Threshold fundamental in the good state and success probability withB1= 3,B2= 2.41,

γ=.085,λ=.5, andp=.005. . . . 52 2.5 Optimal Tobin tax tends to decrease as liquidity shock becomes more severe. . . 52 2.6 Optimal Tobin tax tends to decrease as liquidity shock becomes more frequent. . . 53 2.7 Optimal Tobin tax increases as the volatility of capital ﬂows becomes more harmful. . . . 53 2.8 Maximum probability of success decreases as capital ﬂows volatility becomes more harmful. 54 2.9 The maximum probability of success decreases as liquidity shocks become more severe. . . 54 2.10 The maximum probability of success decreases as liquidity shocks become more frequent. 55

3.1 Decomposition of player 1’s set of choice functions: t∗

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Chapter 1

The Effect of a Tobin Tax on

Coordination in the Presence of

Liquidity Shocks

1.1 Introduction

It has long been argued that a little bit of sand on the well functioning wheels of interna-tional financial markets could be welfare improving. One of the arguments goes like this: long term capital flows are good for the economy, but very short term capital flows might lead to volatility in the capital account and sharp fluctuations in the exchange rate, which is a price that affects the whole economy and moves much faster than prices of goods and labour.1 _{A tax on short term capital flows would then act like a Pigouvian tax.}

Taxes on short term ﬁnancial transactions are often called Tobin tax. In the 1970’s, James Tobin proposed imposing a global tax on all operations involving purchase and sale of foreign currencies as a medium to strength the autonomy of monetary and economic policies of national economies, which, he argued, had been weakened by the increasing mobility and volume of the exchange rates market. Since then, many economists and politicians have advocated the use of ﬁnancial transactions tax on trading of other short

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term securities on the grounds that it would reduce price volatility (see, e.g., Stiglitz 1989 and Summer and Summer 1989), or as a measure of capital control designed to help emerging economies to cope with the risks of ﬁnancial and exchange rate crises due to waves of capital inﬂows (see, e.g., IMF 2012).

However, from a theoretical point of view, the effect of a Tobin tax is far from clear. First, agents might be subject to liquidity shocks, so taxes on short-term capital flows might discourage investment from agents that might need to withdraw their investments and thus lead to a reduction on long-term capital flows. Second, a Tobin tax affects an agent’s behavior not only through its direct effect on payoffs but also indirectly by affecting her expectations about the actions of others. Hence, it affects agents’ expectations about the level and volatility of capital flows, which would have important feedback effects on agents’ payoffs.

This chapter begins our study of the suitability of a Tobin tax in an economy that relies on foreign investors who are subject to liquidity shocks to finance an investment project in the presence of strategic complementarities in the actions of the foreign investors. This first model consists of a 3-period binary action global-game with a continuum of small investors. In the first period, after the observation of a private signal about a fundamental variable that affects the chances of success of the investment project, foreign players decide whether to invest or stay out of the home economy. In the second period, they get another exogenous noisy signal about the fundamental. Also, a fraction λ of the investors is hit by a liquidity shock à la Diamond and Dybvig (1983) and have to consume immediately. Those not affected by the liquidity shock are able to revise their first period decisions. Depending on their choices in the first period, investors facing liquidity restrictions are either forced to withdraw or to stay out of the project. All agents who invest for only one period have to pay a tax levied on short term investments. The project’s outcome and individual payoffs are realized in the last period.

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to the economy. Moreover, the return to investment depends on whether the economy succeeds. This feedback loop creates a coordination problem among foreign investors.

Here we focus on the impacts of the presence of exogenous liquidity restrictions to the performance of a Tobin tax as a policy designed to increase welfare. We examine a mono-tone equilibrium where agents’ decisions in each period are monotonic (non-decreasing) in their signals and the investment project succeeds if and only if the fundamental variable is above a threshold called the critical fundamental. In this case, a tax on short term capitals increases the probability of successful coordination if and only if it decreases the critical fundamental, which is shown to be decreasing in the volume of total investment in the critical fundamental (the critical total investment) and increasing in the volume of capital exit in the critical fundamental (the critical capital exit).

As expected, a tax on short term capitals always decreases capital exit and tends to skew the composition of the total investment to longer maturities through a discouraging impact on both capital outflows and capital inflows in interim periods. However, its effect on the volume of the total investment in the critical fundamental, and hence its effect on the probability of success, depends on whether the agents are subject to liquidity shocks or not. Ifλ = 0, the negative impact of the tax on capital inflows in interim periods is offset by the lower volume of capital outflows and the volume of the critical total investment is not affected by the tax. In this case, a Tobin tax enhances the probability of success.

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Thus, in the presence of liquidity shocks, the impact of a Tobin tax on the probability of success will depend on which eﬀect is dominant: the reduction in critical total invest-ment or the reduction in the critical volume of capital exit. We show that a Tobin tax fosters success incidence only if the liquidity shock is suﬃciently small.

Related Literature. This work is related to a stream of the literature on coordination games with incomplete information, the so called global games, that emphasizes dynamic aspects of the coordination process such as learning and decision adjustments. Dasgupta (2000) develops a model to study the eﬀects of social learning and signaling in a multi-agent sequential decision problem designed to characterize the informational requirements to generate some type of herd behavior in the presence of strategic complementarities. Dasgupta (2007) and Larson (2005) study the implications of the possibility of delay-ing decisions in coordination games. Kovac and Steiner (2013) consider the eﬀects of reversibility in action in a two stage global game.

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There is a fast growing literature studying capital controls from a macroeconomic per-spective. In those models, capital controls are welfare improving owing to a pecuniary externality that aﬀects how much domestic agents can borrow.2 _{There is also research on}

the eﬀect of sudden stops in macroeconomic models along these lines.3 _{These researches}

provide an economic rationale for capital controls but are silent on liquidity and coordi-nation issues. We believe both approaches are important and complement each other.

In our model, the tax on short term capital is not state contingent, i.e, it is the same whether the liquidity shock occurs or not. Besides being analytically convenient, this choice of modeling has some empirical support. Fernandez, Rebucci and Uribe (2013), for example, investigate whether capital controls are used in a prudential fashion. They ﬁnd no evidence that policymakers change capital controls over the business cycle.

The literature on foreign investment in emerging countries provides some evidence on the positive correlation between capital inﬂows and economic success as well as on the negative externality of investment volatility: Oliva and Rivera-Batiz (2002) and de Mello (1997) show that foreign direct investment is associated with growth and economic success, while Lensink and Morrisey (2006) ﬁnd that volatility of foreign direct investment has a negative impact on growth. Previous works have found a detrimental role of economic volatility in general to growth (Ramey and Ramey, 1995; Mobarak, 2005) and welfare (Pallage and Robe, 2003).

Similar results are also found at the microeconomic level: Froot et al (1993) and Minton and Schrand (1999) show that cash ﬂow volatility reduces capital expenditure, investment in R&D, and advertising.

This chapter is organized as follows. The next section describes the model. Section 1.3 shows that it has a unique monotone equilibrium and characterizes it when the agents learn fast. Section 1.4 discusses the consequences of the presence of liquidity shocks to the impact of a Tobin tax on the probability of successfully coordinated investment. Section 1.5 presents our ﬁnal remarks.

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1.2 The model

We consider an economy with 3 periods, indexed by t _{∈ {}1,2,3_}, and a measure 1 of foreign agents, i _∈ [0,1], who make investment decisions at t = 1 and t = 2. Individual payoﬀs are realized at t = 3 and depend on the agent’s action, on players’ aggregate behavior, and on a fundamental variable that describes the state of the economy. Foreign agents are subject to a liquidity shock along the lines of Diamond and Dybvig (1983): a fractionλ ∈[0,1] of them has to consume in the second period. Liquidity needs are only privately observed and uninsurable.

Players make binary decisions, ai

t ∈ {0,1}, at the beginning of the ﬁrst two periods,

i.e, att_{∈ {}1,2_}. At the ﬁrst date, they choose between entering (ai

1 = 1) or not (ai1 = 0)

in an investment project. At t = 2, an agent not aﬀected by the liquidity shock and who invested in the ﬁrst period decides whether to stays in (ai

2 = 1) or leave the project

(ai

2 = 0). Likewise, those who did not invest in the ﬁrst period and were not hit by the

shock can enter (ai

2 = 1) or stay out (ai2 = 0). Depending on their choices att= 1, agents

facing liquidity restrictions are either forced to withdraw or prevented from entering into the project. For convenience, we assume that all agents make a binary decision at t = 2 and interpret ai

2 as the action the i-th player would take if not aﬀected by the liquidity

shock.

Letφi _{be a indicator variable that assumes value 1 when the i-th investor is hit by the}

liquidity shock. Agent i’s payoﬀ depends on her preferred path of actionzi = (ai1, ai2), on

the project outcomeo _{∈ {}0,1_}, which may be a success (o = 1) or a failure (o = 0), and onφi_:

u(zi, φi, o) =



  

  

−τ ai

1, if φi = 1

b(zi₎₍₂_o₋₁₎₋_{τ v}₍_zi₎_, _if _φi _{= 0} (1.1)

whereb(zi_{) =}_ai

1ai2+ai2(1−ai1) is agent i’s ﬁnal investment andv(zi) = ai1(1−ai2)+ai2(1−ai1)

is an indicator for short term capital ﬂows in the path zi_{. According to expression 1.1,}

leaving the project in the interim period or entering only in the second period have a cost

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The project outcome depends on an exogenous fundamental θ, on the volume of in-vestment and on the amount of capital reversals:

o =



  

  

1 if R1

0 d(zi, φi)di≥1−θ

0 if R1

0 d(zi, φi)di <1−θ

(1.2) whered(zi_{, φ}i_{), agent i’s contribution to success, is given by:}

d(zi, φi) = (1₋φi)[b(zi)₋γe(zi)]₋φiγai₁

and e(zi_{) =} _ai

1(1−ai2) indicates if there is capital reversal in the path zi. So, in this

model, capital exit is detrimental to successful investment. The parameter γ ∈ [0,1) is intended to measure in a simple way how harmful capital outﬂows can be to the economy.

1.2.1 Information Structure

The information structure used in this chapter is taken from Mathevet and Steiner (2013). This means that the fundamental θ is a random variable uniformly distributed on the interval [θmin, θmax] and that agents learn about it through the observation of private

signals:

xi

t=xit+1+σηti, t = 1,2 (1.3)

xi₃ =θ (1.4) The errors, ηi

t, are independent across agents and rounds and have continuously

diﬀer-entiable density (f) and distribution (F), and a bounded support [₋η, η]. f is assumed to be bounded below by f > 0, and the support of θ is assumed to contain dominance regions. Also, agents cannot observe their opponents’ actions.

In addition, we assume that individual signals are independent of the probability of an agent being hit by the liquidity shock (λ).

1.2.2 Strategies and Equilibrium

A pure strategys is a family of functions sh :R → {0,1}, h ∈ {∅,0,1}. s∅(x1) indicates

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action she will choose at t = 2 given signal x2 and previous action a1. Since θ and ηt

are bounded, signals will also be bounded. Letx and ¯xbe the minimal and the maximal values of the signalx. We assume thatsh(x) =sh(x) for allx≤x, and thatsh(x) = sh(¯x)

for allx≥x¯. s is said to be a threshold strategy if, for all history h, there existsx∗_h such that sh(x) = 1 for x≥xh∗, andsh(x) = 0 for x < x∗h.

An outcome function O :R_{→ {}₀_,₁_} _{is deﬁned as an indicator of whether the project}

has succeed or not, i.e, O(θ) = o. O is said to be a threshold outcome function if there exists a critical fundamental θ∗ _{such that} _O₍_θ_{) = 1 for}_θ _≥_θ∗_{, and} _O₍_θ_{) = 0 for} _{θ < θ}∗_.

Outcome functions will be extended to the entire real line4 _{assuming that} _O₍_θ_{) = 1 for}

θ_≥θmax, and O(θ) = 0 for θ < θmin.

An strategys will be deﬁned as the best response to an outcome functionO if it is the solution of the following recursive problem

sa1(x2)∈arg max

a′_∈{₀_,₁_}Eθ[b(a1, a

′₎₍₂_O₍_θ₎₋₁₎₋_{τ v}₍_a

1, a′)|x2] (1.5)

s_∅(x1)∈arg max

a′_∈{₀_,₁_}{−λτ a

′_{+ (1}₋_λ₎_E_[_W₍_a′_{, x}

2)|x1]} (1.6)

where

W(a1, x2) = max

a′_∈{₀_,₁_}Eθ[b(a1, a

′₎₍₂_O₍_θ₎₋₁₎₋_{τ v}₍_a

1, a′)|x2] (1.7)

To guarantee that the best response to an outcome function is uniquely deﬁned, it is assumed that agents choose to invest whenever problem 1.5 or problem 1.6 have more than one solution.

We will look for symmetric, pure-strategy, Bayes-Nash equilibria. Let z(x;s) be the preferred investment path of an agent that observe signals x = (x1, x2)T and follows

strategys. If all agents use the same strategys, by the law of large numbers, the aggregate contribution to successR1

0 d(zi, φi)di in state θ equals the conditional expectation of the

individual contribution:

E[d(z(x;s), φi₎_|_θ_{] = (1}₋_λ₎_E_[_b₍_z₍_x_;_s₎₎₋_γe₍_z₍_x_;_s₎₎_|_θ_]₋_λγE_[_eT

1z(x;s)|θ] (1.8)

4As in Mathevet and Steiner (2013), we extend strategies and outcome functions to the real line to

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whereeT

1 = (1,0). An outcome function O will be said to be generated by a strategy s if

O(θ) =



  

  

1, if E[d(z(x;s), φi₎_|_θ_]_≥₁₋_θ

0, if E[d(z(x;s), φi₎_|_θ_]_<₁₋_θ (1.9)

for each θ ∈[θmin, θmax]

An equilibrium is a pair (s, O) such thatsis the best response toO, andOis generated bys. It is a threshold equilibrium if both O and s are threshold functions.

1.2.3 Translational Symmetry

As stressed in Mathevet and Steiner (2013), the uniform prior and the additive errors imply that the joint density of the state θ and the signals x = (x1, x2) is translation

invariant in the interior of its support:

f(θ,x) =f(θ+δ,x+ (δ, δ)T) (1.10) This translational symmetry is inherited by strategies that are best response to threshold outcome functions:

Lemma 1. Let O be a threshold outcome function and suppose that its best response s is a threshold strategy. Then, if O′₍_θ_{) =}_O₍_θ₋_δ₎_{, the best response to} _O′ _{is the strategy} _s′

that satisfies s′_h(x) = sh(x−δ) for all h.

Given any θ∗_{, let} _E_[_d₍_z₍_x_;_s₎_{, φ}i₎_|_θ∗_{] be the aggregate success contribution when the}

fundamental is given byθ∗_{, and all agents follow the strategy}_s _{that is the best response}

to the outcome function

O(θ) =



  

  

1, if θ ≥θ∗

0, if θ < θ∗

(1.11) SinceE[d(z(x;s), φi₎_|_θ_]₋_{1 +}_θ _{is continuous in} _θ_{, we must have}

E[d(z(x;s), φi)|θ∗] = 1−θ∗

Assuming further thatsis a threshold strategy, lemma 1 and the translational symmetry of the joint density f imply that E[d(z(x;s), φi₎_|_θ∗_{] is independent of} _θ∗_{. Thefore, we}

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1.3 Equilibrium Existence and Characterization under Fast

Learn-ing

In this section we establish existence of an threshold equilibrium and characterize it under the assumption that agents’ information is very accurate and increases greatly in each round. More speciﬁcally, it is assumed that

x1 =x2 +ση1 (1.12)

and

x2 =θ+σ2η2 (1.13)

Note that, according to equations 1.12 and 1.13, the ratio of the signal precisions across the two ﬁrst periods diverges as σ _→ 0. This is the fast learning information model of Mathevet and Steiner (2013).

Proposition 3. Suppose that agents receive signals given by equations 1.12 and 1.13 and that λ < 1/2. Then, there exists ¯σ > 0 such that the game has a unique threshold equilibrium for each σ_∈(0,σ¯].

The uniqueness of the threshold equilibrium follows from lemma 2. To prove existence, we build on an argument of Mathevet and Steiner (2013) and show that the best response to a threshold outcome function is a threshold strategy (see lemma 6 in the appendix). Using this result, it is demonstrated that a threshold strategy generates a non decreasing expected success contributionE[d(z(x;s), φ)_|θ] for allσ suﬃciently small, which implies that the project must transit from failure to success at some point θ∗_.

We now characterize the critical state of the threshold equilibrium in the fast learning setting. Given lemma 2, it is suﬃcient that we calculate the limit expected success contribution in the critical state E[d(z(x;s), φ)|θ∗_{]. We do so computing the limiting}

distribution of the investment paths in the critical state asσ →0.

Lemma 4. Let sσ _{be the best response to the outcome function that has threshold} _θ∗ _and

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signals x= (x₁, x₂). If λ <1/2, Then

lim

σ→0P[z(x;s

σ_{) = (1}_,₁₎

|θ∗] = [1 ₋ F(η₁∗)][1 ₋ F(η∗₂₁)] =

"

1₋2λ

2(1−λ)

#

1 +τ

2 (1.14)

lim

σ→0P[z(x;s

σ_{) = (1}_,₀₎

|θ∗] = [1₋F(η∗₁)]F(η₂₁∗ ) =

"

1−2λ

2(1₋λ)

#

1−τ

2 (1.15) lim

σ→0P[z(x;s

σ_{) = (0}_,₁₎_|_θ∗_{] =}_F₍_η∗

1)[1−F(η20∗ )] =

"

1 2(1₋λ)

#

1−τ

2 (1.16)

and

lim

σ→0P[z(x;s

σ_{) = (0}_,₀₎

|θ∗] =F(η∗₁)F(η∗₂₀) =

"

1 2(1−λ)

#

1 +τ

2 (1.17)

where η∗₂_a₁ = limσ→0

x∗

2a₁(σ,τ,λ)−θ∗

σ2 , η∗₁ = limσ→0 x ∗

1(σ,τ,λ)−θ∗

σ , andx∗1(σ, τ, λ) and x∗2a1(σ, τ, λ) are the threshold signals that defines the strategy sσ_.

In these equations, F(η₁∗) represents the lower bound on the probability of success at

t = 1 necessary for an agent to be willing to invest as σ → 0, and F(η₂∗_a₁) indicates the success probability required by an agent with history h=a1 to invest at t= 2 as σ→0.

When an agent observes signalxt at roundt she updates her belief about the probability

of success to qt(xt) = P r[θ ≥ θ∗|xt] and chooses at = 1 only if she believes that the

probability of success is greater or equal to F(η∗

th). The uniform prior distribution of θ implies that the distribution of the posterior beliefs at the critical state θ∗ _{is uniform}

on [0,1], i.e, qt(xt)|θ∗ ∼ U[0,1] (see Guimaraes and Morris (2007), Steiner (2006), and

Mathevet and Steiner (2013)). So, the mass of agents who choose to invest at round t is given by 1−F(η∗_th), where h =∅ when t = 1. In addition, under the assumption of fast learning, the posterior beliefs becomes independent across rounds as σ → 0. Therefore, ignoring the liquidity shock, the probability of observing, say, a player investing at both rounds in the critical state (P[z(x;sσ_{) = (1}_,₁₎_|_θ∗_{]) converges to [1}₋_F₍_η∗

1)][1−F(η∗21)] as

σ_→0. Given

e(z) =a1(1−a2) =

      

1, if z = (1,0)

0, if z _{∈ {}(0,0),(0,1),(1,1)_}

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we have

E[e(z(x;sσ))|θ∗] =X z

es(z)P[z(x;sσ) =z|θ∗] =P[z(x;sσ) = (1,0)|θ∗] (1.19) Thus, the limit expected value of the voluntary capital outﬂow is given by

lim

σ→0E[e(z(x;s

σ₎₎

|θ∗] = lim

σ→0P[z(x;s

σ_{) = (1}_,₀₎

|θ∗] =

"

1₋2λ

2(1−λ)

#

1₋τ

2 (1.20) Also,

eT1z =

      

1, if z∈ {(1,0),(1,1)}

0, if z_{∈ {}(0,0),(0,1)_}

(1.21) Therefore,

E[eT1(z(x;sσ))|θ∗] =P[z(x:sσ) = (1,0)|θ∗] +P[z(x:sσ) = (1,1)|θ∗] (1.22)

equals the probability of investing in the ﬁrst period, and has limit lim

σ→0E[e

T

1(z(x;sσ))|θ∗] =

"

1₋2λ

2(1₋λ)

#

(1.23) The limit volume investment in the critical state is given by

lim

σ→0(1−λ)E[b(z(x;s

σ₎₎

|θ∗] = lim

σ→0(1−λ)[P[z(x:s

σ_{) = (1}_,₁₎

|θ∗] +P[z(x:sσ) = (0,1)_|θ∗]] = (1−λ)

"

1 2−

λτ

2(1₋λ)

#

(1.24) Substituting equations 1.20, 1.23 and 1.24 into equation 1.8, it follows from lemma 2 that the equilibrium critical state asσ _→0 is

θ∗ = 1₋(1₋λ)

"

1 2−

τ λ

2(1−λ)

#

+ (1₋λ)γ "

1₋2λ

2(1−λ)

#

1₋τ

2 +λγ

"

1₋2λ

2(1−λ)

#

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aﬀected by the liquidity shock is smaller than the negative impact of capital outﬂows. More precisely,

Proposition 5. A Tobin tax increases the ex-ante probability of success if and only if

λ < γ

2(1 +γ) (1.26)

Proof: Notice that the tax increases the probability of success only if it decreases the critical fundamental θ∗. From quation 1.25, we have

dθ∗

dτ =

λ

2 −

γ

2

"

1₋2λ

2

#

(1.27) Thus,

dθ∗

dτ <0⇔λ < γ

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1.4 The Impact of a Liquidity Shock

To understand the consequences of the presence of liquidity shocks to the eﬀectiveness of a Tobin tax as a policy designed to increase success incidence and welfare, this section highlights the distinctions between the casesλ =φi _≡_{0 and}_{λ >}_0.

According to equation 1.24, the volume of investment in the critical state does not depend on the tax when λ = 0. This is a speciﬁc case of a more general result due to Mathevet and Steiner (2013) known as the invariance of the critical investment. Under the assumption thatλ=φi _≡_{0, the payoﬀ function 1.1 reduces to ˜}_u₍_{z, o}_{) = (2}_o₋₁₎_b₍_z₎₋ τ v(z), and it can be written as

˜

u(z, o) = ˜b(z)o−c(z) (1.29) where ˜b(z) = 2b(z) and c(z) = b(z) +τ v(z). Given the information structure in section 1.2.1 and a payoﬀ function of the form 1.29, the invariance result states that the volume of investment in the critical state, b∗_{, satisﬁes}

2b∗ _≡˜b∗ = max

z u˜(z,1)−maxz u˜(z,0) (1.30)

Thus, frictions that do not aﬀect maxzu˜(z,1) or maxzu˜(z,0) cannot impact the critical

volume of investment. In our speciﬁc case, we have maxzu˜(z,1) = ˜u((1,1),1) = 1 and

maxzu˜(z,0) = ˜u((0,0),0) = 0; that is, an informed agent who knows that the project will

succeed enters in the ﬁrst period and never leaves the economy, and a player who knows that the project will fail never invests. Since a Tobin tax is levied only on short term capitals, it will not aﬀect the critical investment in the absence of liquidity shocks.

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Another consequence of assuming that players may face liquidity restrictions is to create conditions for the existence of a harmful eﬀect of a Tobin tax on success incidence. The following discussion shows the intuition behind this result.

The critical fundamental can be written as

θ∗ = 1−ˆb∗ +γeˆ∗ (1.31) where ˆb∗ = (1−λ)b∗ stands for the total investment in the critical state, ˆe∗ = [(1−λ)e∗+

λp∗

1] indicates the volume of capital outﬂow in the critical state and p∗1 represents the

mass of players who invest at t = 1 in the critical state. So, a decreasing critical total investment tends to require stronger fundamentals in order to achieve economic success.

To understand why the critical total investment can be a decreasing function of the tax, notice that it can be decomposed as the sum of the mass of players who choose to invest att = 1 and stay at t = 2 (long-run investment) and the mass of those who enter in the project at t = 2 (short term investment). In our model, a Tobin tax stimulates long run investment and discourages short term investment. Economically, the decreasing critical total investment and the possibility of observing an adverse eﬀect of a Tobin tax on the probability of success will be then driven by the fact that a liquidity shock weakens the tax’s capacity of promoting long run investment without reducing its discouraging eﬀect on short term investment. To see this, let us consider the cases without and with liquidity shock in turn.

Assume λ = 0 ﬁrst. In this case, by equations 1.14 and 1.16, the critical long term investment (L∗

0) and the critical short term investment (S0∗) are given by

L∗₀ = 1 2

1 +τ

2

(1.32) and

S₀∗ = 1 2

1−τ

2

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tax. These results are illustrated in ﬁgure 1.1, which gives the probabilities of playing actionat, conditional on reaching historyh, for the case without liquidity restrictions.

Now consider the case λ >0. Given that the probability of being hit by the liquidity shock is independent among the players, using equations 1.14-1.17, the conditional mass of capital ﬂows, given node h _{∈ {∅},0,1_} and the liquidity shock, are depicted in ﬁgure 1.2. The critical long term investment (L∗

λ) and the critical short term investment (Sλ∗)

are now given by

L∗_λ =

"

1₋2λ

2(1₋λ)

#

(1₋λ)(1 +τ) 2 =

(1₋2λ) 2

(1 +τ)

2 (1.34) and

S_λ∗ =

"

1 2(1−λ)

#

(1₋λ)(1₋τ) 2 =

1 2

(1₋τ)

2 (1.35) Note that a Tobin tax still encourages long-term investment. However, this impact is weakened by the liquidity shock. First, compared with the case without liquidity shock, there is a fall in the mass of players who invest att = 1. The possibility of being hit by a liquidity shock induces players to delay their decisions. Second, only a fraction (1₋λ) of those who believe att= 2 that the project will succeed, which, as in the case withλ = 0, is given by (1+τ)/2, are not hit by the liquidity shock and can stay in the economy. These two eﬀects contribute to make long term investment decreasing in the liquidity shock and reduces the marginal gains in critical long run investment due to the adoption of a Tobin tax by λ/2.

The eﬀect of the liquidity shock on the critical short term investment is less clear cut. Short term investment is given by a fraction of the mass of investors who choose to stay out of the project at t = 1. A liquidity shock increases the mass of agents that stay out att = 1, but decreases the mass of players that are able to enter in the economy att = 2. Here, these eﬀects cancel each other out in the critical state and the critical short term investment becomes invariant to liquidity restrictions.

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0

a2= 0 1+τ

2

1 a2= 1

1−τ 2

a1= 0

1/2

1

a2= 1 1+τ

2

0 a2= 0

1−τ 2

a1= 1

1/2

b∗= 1 2

e∗= 1−₄τ

Figure 1.1

Capital flows with λ= 0

Probabilities of choosing actionatin the critical stateθ∗, conditional on reaching historyh∈ {∅,0,1}in

the absence of liquidity shock. The variablesb∗ _and_e∗_{indicate the critical investment and the critical}

volume of capital exit, respectively.

Now, consider the conditions that are required for a Tobin tax to enhance success incidence. Using equations 1.20 and 1.23, we found that the critical capital outﬂow is given by

ˆ

e∗ = 1−2λ 2

(1−τ) 2 +

λ(1−2λ)

2(1₋λ) (1.36) So, the marginal decrease in capital exit due to the adoption of a Tobin tax is given by (1−

2λ)/4. From equation 1.31, a Tobin tax decreases the critical state, and hence increases theex-ante probability of success, if and only if the marginal loss in total investment does not surpass the weighted marginal decrease in capital exit. Numerically, this is equivalent toλ/2< γ(1₋2λ)/4, which is exactly the condition derived in proposition 5, i.e,

λ < γ

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0 (1−λ)(1+τ)

2 +λ

1

(1−λ)(1−τ) 2 1 2(1−λ)

1

(1−λ)(1+τ) 2

0

(1−λ)(1−τ)

2 +λ

1−2λ 2(1−λ)

˜_b∗₌ 1

2− τ λ 2(1₋λ)

˜

e∗= 1−2λ 2

(1₋τ)

2 +

λ(1₋2λ) 2(1₋λ)

Figure 1.2 Capital flows with λ >0

Mass of capital ﬂows in the critical stateθ∗, conditional on reaching historyh∈ {∅,0,1}in the case of a positive liquidity shock. The variablesb∗ ande∗indicates the critical total investment and the critical

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1.5 Conclusion

This chapter studies the consequences of the presence of liquidity shocks to the perfor-mance of a Tobin tax as a measure designed to increase success incidence in a dynamic global game setting where coordination success is increasing in the volume of foreign di-rect investment, but decreasing in the volume of capital outflow. We find that liquidity restrictions may harm the tax’s capacity of promoting long term investment more than they stimulate investment in interim periods (short term investment). In this case, a Tobin tax compromises total investment and may decreases the chances of success if the economy does not benefit sufficiently from a reduction on capital exit.

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1.6 Appendix

1.6.1 Proof of Lemma 1

Lets be the best response to the outcome function

O(θ) =



  

  

1 if θ _≥θ∗

0 if θ < θ∗ (1.37)

The incentive to invest at t= 2 given action a1 att = 1 and signal x2 is given by

Πσ,τ,λ_a₁ (x2) =E[2b(a1,1)O(θ)−c(a1,1)|x2]−E[2b(a1,0)O(θ)−c(a1,0)|x2] (1.38)

wherec(z) =b(z) +τ v(z). Thus,

[2b(a1,1)O(θ)−c(a1,1)|x2]−[2b(a1,0)O(θ)−c(a1,0)|x2]

= 2O(θ)₋1₋τ(1₋a1) +τ a1

= 2O(θ)−(1 +τ) + 2τ a1 (1.39)

Therefore,

Πσ,τ,λ_a₁ (x2) = 2E[O(θ)|x2]−(1 +τ) + 2τ a1

= 2P [θ _≥θ∗_|x2]−(1 +τ) + 2τ a1

= 2F2

x2−θ∗

σ2

!

−(1 +τ) + 2τ a1 (1.40)

Given thatF2 is strictly increasing inx2, Πaσ,τ,λ1 (x2) will be strictly increasing inx2. Hence,

the threshold deﬁningsa1 must be the unique value x∗2,a1(θ

∗_{) that satisﬁes}

Πσ,τ,λ_a₁ (x∗₂_,a₁(θ∗)) = 0 (1.41)

Moreover, Πσ,τ,λ

a1 (x2) strictly increasing implies that sa1 will be a threshold function for all

strategysthat is a best response to a threshold outcome function. Also, given a threshold fundamentalθ′ ₌_θ∗₊_δ_{, condition 1.41 can only be satisﬁed by} _x∗

2,a1(θ

′_{) =}_x∗

2,a1(θ

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Now, consider s_∅. We have

s_∅(x1)∈arg max

a′

∈{0,1}{−λτ a

′_{+ (1}₋_λ₎_E_[_W₍_a′_{, x}

2)|x1]} (1.42)

where

W(a1, x2) = max

a′_∈{₀_,₁_}E[2b(a1, a

′₎_O₍_θ₎₋_c₍_a

1, a′)|x2]

When a′ _{= 1, we have}

−λτ a′+ (1−λ)E[W(a′, x2)|x1]

=₋λτ + (1₋λ)E[W(1, x2)|x1]

=₋λτ + (1₋λ)E "

2F x2−θ∗ σ

!

−1

!

I_{x2≥x∗₂₁}(x2)|x1 #

−(1−λ)τ E[(1−I_{x2≥x∗21}(x2))|x1]

=₋λτ + (1₋λ)E "

2F x2−θ∗ σ

!

−1

!

I_{x2≥x∗₂₁}(x2)|x1 #

−(1₋λ)τ(1₋P[x2 ≥x∗21|x1])

=−τ+ 2(1−λ)

Z ∞ x∗

21

F x2−θ

∗

σ !

f x

1−x2

σ

1

σdx2

−(1₋λ)(1₋τ)F x

1−x∗21

σ

(1.43) with a′ = 0, it follows that

(1₋λ)E[W(0, x2)|x1] =

(1₋λ)(1 +τ)F x

1−x∗20

σ

+ 2(1₋λ)

Z ∞ x∗

20

F x2−θ∗ σ

!

f x

1−x2

σ

1

σdx2 (1.44)

So, the thresholdx∗

1 must satisfy

2

Z x∗₂₀

x∗

21

F x2−θ∗ σ2

!

f x∗

1−x2

σ

1

σdx2

=−(1−τ)F x∗

1−x∗21

σ

+ (1 +τ)F x∗

1−x∗20

σ

+ τ

1₋λ (1.45)

In this case, x′1 = x∗1 +δ will also satisfy equality 1.45 when θ′ = θ∗ +δ, x′21 = x∗21+δ

and x′

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between the thresholds. So, the right hand side of 1.45 does not change. There is no change in the left hand side either because

Z x∗₂₀+δ

x∗₂₁+δ F

x2−θ∗−δ

σ2

!

f x∗1+δ−x2 σ

!

1

σdx2

=

Z x∗₂₀

x∗

21

F x2+δ−θ∗−δ σ2

!

f x∗1+δ−x2−δ σ

!

1

σdx2

=

Z x∗₂₀

x∗

21

F x2 −θ

∗

σ2

!

f x∗

1−x2

σ

1

σdx2

(1.46) Finally, since thatsis a threshold strategy, the incentive to invest given outcome function

O must be positive for x1 > x∗1 +δ and negative for x1 < x∗1 +δ. Hence, the strategy

characterized by the thresholdsx∗₂_,a₁(θ′) = x∗₂_,a₁(θ∗) +δ and x′₁ =x∗₁+δ must be the best response to the threshold outcome function O′ whose threshold fundamental is given by

θ′ ₌_θ∗₊_δ_.

1.6.2 Proofs for section 1.3

Lemma 6. Suppose that the signals satisfy the fast learning assumption and thatλ <1/2. Then there exists σ >¯ 0 such that the best response to a threshold outcome function is a threshold strategy for each σ_∈(0,¯σ].

Proof:

Suppose that the investment project is successful only if θ _≥ θ∗ _{and let} _s _{be its best}

response. It was shown In the proof of lemma 1 that the functions sa1 have thresholds x∗

2,a1(θ

∗_{) for all} _σ_{. So, it is suﬃcient to proof that} _s

∅ is a threshold function. Let

π_∅σ(x1) = {−λτ + (1−λ)E[W(1, x2)|x1]} −(1−λ)E[W(0, x2)|x1]}

={−λτ + (1−λ)E[W(1, x2)−W(0, x2)|x1]} (1.47)

denote the incentive to invest of a player that observes signalx1 att = 1.

Given that ηt has support in [−η, η] and x2 = θ +σ2η2, If x2 > θ∗ +σ2η, then θ > θ∗,

i.e, the agent will be certain at t = 2 that the project will succeed. Analogously, If

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the agent will be certain of the success of the project att= 2, conditional x1, is

P rσ[x2 > θ∗+σ2η|x1] =P[x1−ση1 > θ∗+σ2η|x1] =

P "

η1 <

x1 −θ∗−σ2η

σ

#

=F x1−θ

∗₋_σ2_η

σ

!

(1.48) The probability that, att= 2, the agent will be certain that the project will fail, coditional

x1, is

P rσ[x2 < θ∗−σ2η|x1] =P[x1−ση1 < θ∗−σ2η|x1] =

P "

η1 ≥

x1−θ∗+σ2η

σ

#

= 1₋F(x1−θ∗+σ

2_η

σ ) (1.49)

Givenx1 =x2+ση1 =θ+σ2η2+ση1, it follows that the support ofx1 is given by [x1,x¯1],

wherex1 =θmin−η(σ+σ2) and ¯x1 =θmax+η(σ+σ2). We will investigate the behavior

ofπσ

∅(x1) in the three following subintervals of the support of x1: I1 = [x1, θ∗+σ2η−ση],

I2 = [θ∗+σ2η−ση, θ∗−σ2η+ση] and I3 = [θ∗−σ2η+ση,x¯1].

Consider x1 ∈I3. In this case, −2σ2_η₊_ση

σ ≤

x1−θ∗−σ2η

σ ≤

¯

θ₋θ∗₊_ση

σ

Therefore,

lim

σ→0F

x1 −θ∗−σ2η

σ

!

= 1 _∀ x1 ∈I3 (1.50)

So, the probability that the agent will be certain of the project success att= 2, givenx1,

converges uniformly to 1 in the intervalI3 as σ→0.

By deﬁnition, we have5

W(1, x2) = max

a′_∈{₀_,₁_}E[2b(1, a

′₎_O₍_θ₎₋_c₍₁_{, a}′₎_|_x

2]

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where

E[2b(1,1)O(θ)₋c(1,1)_|x2] =E[2O(θ)−1|x2]

= 2E[O(θ)|x2]−1 = 2P[θ ≥θ∗|x2]−1 = 2F

x2−θ∗

σ2

!

−1 (1.51) and

E[2b(1,0)O(θ)₋c(1,0)_|x2] =−c(1,0) =−τ

Letx2(σ) be the uniquex2 that satisﬁes

2F x2(σ)−θ∗ σ2

!

= 1₋τ

Then, we can write

W(1, x2) =

"

2F x2 −θ

∗

σ2

!

−1

#

I_{x2≥x2(σ)}(x2)−τ h

1−I_{x2≥x2(σ)}(x2) i

Therefore,

E[W(1, x2)|x1] =

E ""

2F x2−θ∗ σ2

!

−1

#

I_{x2≥x2(σ)}(x2)−τ h

1₋I_{x2≥x2(σ)}(x2) i

|x1

#

(1.52) Notice thath2F x2−θ∗

σ2

−1iI_{x2≥x2(σ)}(x2) andI{x2≥x2(σ)}(x2) are non-negative functions

that converge monotonically toI_{x2≥θ∗}(x2) as σ→0. Hence, for all x1 ∈I3,

lim

σ→0E[W(1, x2)|x1] =E

h

I_{x2≥θ∗}(x2)|x1 i

= 1 (1.53) where the last equality follows from expression (1.50).

Now consider

W(0, x2) = max

a′_∈{₀_,₁_}E[2b(0, a

′₎_O₍_θ₎₋_c₍₀_{, a}′₎_|_x

2]

We have

E[2b(0,1)O(θ)−c(0,1)|x2] =E[2O(θ)−(1 +τ)|x2] =

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and

E[2b(0,0)O(θ)₋c(0,0)_|x2] = 0 (1.55)

Therefore, using expression (1.50), it follows that lim

σ→0E[W(0, x2)|x1] = 2−(1 +τ) = 1−τ (1.56)

for all x1 ∈I3. Combining equations 1.47, 1.53 and 1.56 we ﬁnd that

lim

σ→0π

σ,τ,λ

∅ (x1) = −τ λ+ (1−λ)[1−(1−τ)]

=−τ λ+ (1−λ)τ =−2λτ +τ (1.57) Given that λ < 1/2, it follows that π_∅σ,τ,λ(x1) > 0 for all x1 ∈ I3 provided that σ is

suﬃciently small.

Next, consider x1 from the interval I1. The expression in 1.49 converges uniformly to

1 onI1 asσ →0. In this case, a similar argument gives limσ→0E[W(1, x2)|x1] =−τ and

limσ→0E[W(0, x2)|x1] = 0. Therefore,

lim

σ→0π

σ,τ,λ

∅ (x1) = −τ λ+ (1−λ)[E[W(1, x2)|x1]−E[W(0, x2)|x1]]

=−τ λ+ (1−λ)(−τ)<0 (1.58) So, πσ,τ,λ_∅ (x1)<0 for all x1 ∈I1 provided that σ is suﬃciently small.

Now, assume that x1 ∈I2. Since an agent who observes signal

x2 < θ∗−σ2η knows that the project will fail, we have

W(0, x2) = max

a′_∈{₀_,₁_}E[2b(0, a

′₎_O₍_θ₎₋_c₍₀_{, a}′₎_|_x

2] =−c(0,0) = 0

and

W(1, x2) = max

a′

∈{0,1}E[2b(1, a

′₎_O₍_θ₎₋_c₍₁_{, a}′₎_|_x

2] = max

a′

∈{0,1}−c(1, a

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Analogously, givenx2 > θ∗+σ2η, we have

W(0, x2) = max

a′_∈{₀_,₁_}E[2b(0, a

′₎_O₍_θ₎₋_c₍₀_{, a}′₎_|_x

2]

= max

a′_∈{₀_,₁_}[2b(0, a

′₎₋_c₍₀_{, a}′_)]

= max_{2b(0,1)₋c(0,1),0_}= 2b(0,1)₋c(0,1) = 1₋τ (1.59) and

W(1, x2) = max

a′_∈{₀_,₁_}E[2b(1, a

′₎_O₍_θ₎₋_c₍₁_{, a}′₎_|_x

2]

= max

a′_∈{₀_,₁_}{2b(1, a

′₎₋_c₍₁_{, a}′₎_}_{= max}_{₂_b₍₁_,₁₎₋_c₍₁_,₁₎_,₂_b₍₁_,₀₎₋_c₍₁_,₀₎_}

= max{1,−τ}= 1 (1.60) So, the incentive to invest att = 1 can be written as

π_∅σ,τ,λ(x1) =−λτ + (1−λ)

Z

[W(1, x2)−W(0, x2)]f(x2|x1)dx2

=₋λτ+ (1₋λ)[

Z θ∗−σ2η

−∞ [W(1, x2)−W(0, x2)]f(x2|x1)dx2+

Z θ∗+σ2η

θ∗₋_σ2_η [W(1, x2)−w(0, x2)]f(x2|x1)dx2

+

Z ∞

θ∗₊_σ2_η[W(1, x2)−W(0, x2)]f(x2|x1)dx2]

=₋λτ + (1₋λ)[(₋c(1,0) +c(0,0))P[x2 ≤θ∗−σ2η|x1]

+

Z θ∗+σ2η

θ∗₋_σ2_η [W(1, x2)−w(0, x2)]f(x2|x1)dx2

+ (2b(1,1)₋c(1,1)₋2b(0,1) +c(0,1))P[x2 ≥θ∗ +σ2η|x1]] (1.61)

LetM be the real function given by

M(x2) =



  

  

W(1, x2)−W(0, x2) + [c(1,0)−c(0,0)], se x2 < θ∗

W(1, x2)−W(0, x2) + [(2b(1,1)−c(1,1))−(2b(0,1)−c(0,1))]

Flexible information acquisition and optimal Tobin tax in tractable dynamic global games

Contents

List of Tables

List of Figures

Chapter 1

The Effect of a Tobin Tax on

Coordination in the Presence of

Liquidity Shocks

1.1

Introduction

1.2

The model

1.3

Equilibrium Existence and Characterization under Fast

Learn-ing

1.4

The Impact of a Liquidity Shock

1.5

Conclusion

1.6

Appendix