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Essays on government intervention in financial crises

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Essays on government intervention in financial crises

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Content Essays on Government Intervention in Financial Crises by Danilo Lopomo Beteto A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements of the Degree DOCTOR OF PHILOSOPHY (Economics) August 2013 Copyright 2013 Danilo Lopomo Beteto In memory of my grandmother Francisca and to my parents Miriam and Danilo... ii Acknowledgements I am very grateful to my advisor, Michael Magill, and to Vincenzo Quadrini and Fernando Zapatero, members of the committee, for all the help and patience during the execution of this project. I also would like to say thanks to my friends Carlos Bueno Raymundo, Doruk Iris, TymurGabunyia, LuizFichtner, RodrigoAraujo, Daniel Bergmann, VladRadoias and Jaime Meza. Last but not least to my family, Miriam, Danilo, Daniela e Alinne, for always being there when I need, and to Luise Wegner, who inspires me every day. iii Contents Dedication ii Acknowledgements iii List of Tables vii List of Figures viii Abstract ix Chapter 1 Government Intervention and Financial Fragility 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.2 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.1.3 Contributions of the Paper . . . . . . . . . . . . . . . . . . . . . . . 14 1.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.2.1 Banks Interaction Process and Arrival of Depositors . . . . . . . . . 18 1.2.2 Network Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.2.3 Maturity Mismatch . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.2.4 Government Intervention . . . . . . . . . . . . . . . . . . . . . . . . 22 1.2.5 Timeline of Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.3 Link Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.3.1 Investment in a Small Project . . . . . . . . . . . . . . . . . . . . . . 24 1.3.2 Investment in a Large Project . . . . . . . . . . . . . . . . . . . . . . 26 1.3.3 Investment in Loans . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.4 Banks’ Portfolio Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.4.1 Government Intervention and Network Structure . . . . . . . . . . . 30 1.5 Characterization of the Financial System and Shocks . . . . . . . . . . . . . 31 1.6 Measures of Financial Fragility . . . . . . . . . . . . . . . . . . . . . . . . . 39 1.6.1 A Simulation-Based Google Measure of Fragility . . . . . . . . . . . 39 1.6.2 Financial Fragility and Distribution of Parameters . . . . . . . . . . 44 1.6.3 Simulations and Financial Development . . . . . . . . . . . . . . . . 46 1.6.4 Qualitative Results from Simulations . . . . . . . . . . . . . . . . . . 51 1.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 iv Chapter 2 Government Safety Net, Stock Market Participation, and As- set Prices 57 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.1.1 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 2.3 Government Intervention. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 2.3.1 Imperfect Information About the Fundamentals . . . . . . . . . . . . 74 2.3.2 Perfect Information. . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 2.3.3 No Information: Common Prior About the Fundamentals . . . . . . 90 2.3.4 Analysis of the Equilibrium Price Across Different Informational Sce- narios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 2.4 No Government Intervention . . . . . . . . . . . . . . . . . . . . . . . . . . 97 2.4.1 Imperfect Information About the Fundamentals . . . . . . . . . . . . 98 2.4.2 Perfect Information About the Fundamentals . . . . . . . . . . . . . 99 2.4.3 No Information: Common Prior About the Fundamentals . . . . . . 101 2.4.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 2.5 Government vs No Government Prices . . . . . . . . . . . . . . . . . . . . . 104 2.5.1 Imperfect Information . . . . . . . . . . . . . . . . . . . . . . . . . . 105 2.5.2 Perfect Information. . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 2.5.3 Common Prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 2.5.4 Informational Scenarios Combined: the Main Result . . . . . . . . . 107 2.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Chapter 3 Government Induced Bubbles 113 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 3.1.1 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 3.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 3.2.1 Perfect information about the fundamentals . . . . . . . . . . . . . . 126 3.2.2 Imperfect information . . . . . . . . . . . . . . . . . . . . . . . . . . 131 3.3 Comparative statics and policy implications . . . . . . . . . . . . . . . . . . 141 3.3.1 Changes in the information structure . . . . . . . . . . . . . . . . . . 141 3.3.2 Changes in transaction costs . . . . . . . . . . . . . . . . . . . . . . 145 3.3.3 Changes in aggregate wealth . . . . . . . . . . . . . . . . . . . . . . 146 3.3.4 Changes in the threshold level of bubble burst, L . . . . . . . . . . . 147 3.3.5 Changes in the threshold level of government intervention, c . . . . . 148 3.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Bibliography 153 Appendices 159 Chapter A Appendix to Chapter 1 160 A.1 Proof of Proposition 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 A.2 Proof of Proposition 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 A.3 Proof of Corollary 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 A.4 Proof of Corollary 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 A.5 Proof of Lemma 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 v Chapter B Appendix to Chapter 2 171 B.1 Comparison of EquilibriumPrices across Different Frameworks with Govern- ment Intervention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 B.2 Equilibrium Conditions in the Imperfect Information Case without Govern- ment Intervention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 B.3 ComparisonofEquilibriumPricesacrossDifferent Frameworks withoutGov- ernment Intervention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 B.4 Government vs No Government Prices in the Common Prior Scenario . . . 184 vi List of Tables 1.1 Parameters used for the generation of the networks in Figure 1.7. . . . . . . 41 1.2 Ranking of banks according to the number of failures induced. . . . . . . . 42 1.3 Percentual over the number of indirect, direct and total failures. . . . . . . 43 1.4 Parameters used in the simulations performed. . . . . . . . . . . . . . . . . 49 2.1 Asset price characterization in terms of uncertainty and supply level. . . . . 82 vii List of Figures 1.1 Network Structure and Government Intervention. . . . . . . . . . . . . . . . 9 1.2 Region i and its continuums of depositors. . . . . . . . . . . . . . . . . . . . 17 1.3 Cash flows of region i’s projects. . . . . . . . . . . . . . . . . . . . . . . . . 17 1.4 Interaction process of banks for N =4. . . . . . . . . . . . . . . . . . . . . . 18 1.5 Portfolio decision of banks at a particular round of interaction. . . . . . . . 20 1.6 Examples of networks at the end of the interaction process (N =4). . . . . 21 1.7 Networks generated under the parameters of Table 1.1. . . . . . . . . . . . . 41 2.1 Impact of government intervention and feedback. . . . . . . . . . . . . . . . 59 2.2 Effects from government intervention. . . . . . . . . . . . . . . . . . . . . . 63 2.3 Timeline of events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.4 Level of uncertainty as a function of θ ∗ . . . . . . . . . . . . . . . . . . . . . 80 2.5 Equilibrium process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 2.6 Downside-risk intervals and equilibrium price accordi ng to uncertainty and supply levels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 2.7 Support of e θ, distribution of signals, range uncertainty and location of θ ∗ . . 85 2.8 Equilibriumpriceorderingacrossdifferentinformationalscenariosforthelow uncertainty case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 2.9 Equilibrium price ordering across different informational scenarios for the high uncertainty case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 2.10 Equilibrium price ordering across different informational scenarios without the possibility of government intervention. . . . . . . . . . . . . . . . . . . . 104 3.1 Bubble function for α=0 and α =1. . . . . . . . . . . . . . . . . . . . . . . 151 3.2 Mass of speculators buying the asset and critical mass requiring government intervention. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 viii Abstract This thesis studies the effects of government intervention in financial crises. In the first chapter it is analysed the effects of intervention for the formation of the network of banks, whereas in chapters 2 and 3 it is analysed the effects of intervention on asset prices. Chapter is divided in two parts. In the first one, government intervention is defined as a policy that allows banks to reduce fire sale costs, and an endogenous model of the formation of the network of banks is developed in a way that this type of intervention plays a key rolein determiningthe structureof thebankingsystem. Theidea is that intervention brings incentives for banks to invest in more profitable but less liquid assets, which in turn can be financed only through interbank loans, which creates links across banks. In the second part of the paper, financial fragility is defined as the number of bank failures after banks’assetsarehitbyshocks,andunderthismeasurenetworksobtainedwithandwithout intervention are compared. Theoretically it is shown that intervention makes the network of banks to be more connected, and from simulations that it brings more financial fragility despite increasing the wealth of the banking system. Chapter 2 studies the effect of a policy whereby intervention occurs only in financial crises where the welfare of investors goes below a specific threshold, to be called the safety net. The main question is how equilibrium prices differ under different intervention poli- cies, andforthat itis studiedtheinvestment decision problemof agents undertwo different frameworks, with and without intervention, and under three different informational scenar- ios, imperfect information, perfect information, and common priors. These informational scenariosaremeanttocapturedifferentclassesofassetswhereinvestorshavevaryingdegrees of knowledge regarding the technology embedded in the asset. It is shown that, regardless of the informational scenario, equilibrium prices can be sustained at a higher level in the ix framework with intervention than in the one without. However, equilibrium prices cannot be sustained at too much of high levels because, even though that would signal to investors that intervention would take place for sure, it still would not make investments profitable and, therefore, agents would prefer not to participate in the market, making the market clearing condition to fail to hold and an equilibrium not to exist. Chapter 3 studies the effects of a policy such that intervention occurs only in those financial crises where there is a significant drop in the level of prices. It is assumed that everyone knowsthe observed priceof aparticular asset, butonly thegovernment knows the fundamental price, the difference between the two being called a bubble. Nature defines if there will be a crisis or not and, everytime there is one, the bubblebursts and the observed price drops to the fundamental. At this moment the government faces the cost of letting the bubble burst - proportional to the size of the bubble - and the cost of intervention, and by trading off the two it decides on its action. It is shown that, under such a policy rule, investors have incentives to inflate bubbles because by doing so they can make a higher capital gain if there is no crisis and, in case there is one, they increase the likelihood of intervention. x Chapter 1 Government Intervention and Financial Fragility This chapter studies a model in which banks decide on the projects in which they invest, and the banks to which or from which they obtain loans. Thus, the links (network) created between banks is endogenous. Each bank is characterized by parameters which define the return on its projects, the withdrawal rate of its depositors and its equity available for investments. Maturity mismatch of balance sheets forces a fraction of assets to be prematurely liquidated, at a fire sale cost. The paper focuses on the impact of government intervention, whichalleviatesthiscostbyincreasingtherecoveryrateofassets. Thefragility of a network is measured by the numberof bank failures following shocks of two kinds: first a shock to a single bank, second a simultaneous shock to all banks. The first leads to a rankingof thebankssimilar tothatusedbyGoogle torankwebsites: thehigherits ranking thegreaterthedegreeofvulnerabilityinducedbythebank. Thevulnerabilityofthenetwork to simultaneous shocks dependson theprobability distributionof the bankscharacteristics: the more dispersed the distribution the greater its vulnerability. Government intervention increases the vulnerability of the network, the increase being greater the more dispersed 1 the characteristics of the banks. Banking systems with similar leverage can have different degrees of vulnerability, highlighting the importance of networks. 1.1 Introduction The financial crisis of 2007-2009 brought to the fore the ques tion of what should be the role of the government in times of distress. On the one side there were those advocating in favor of an intervention - a bailout of financial instituti ons - so that a collapse of the financial system could be avoided, whereas on the other those claiming that banks should beletattheirownfate, sincetheirfailurewasnothingelsethanaresultofareckless, greedy behavior, inway thatabailoutwoulddonothingbutexacerbate themoralhazardproblem. What makes the government sensitive to the bailout of a particular institution is the risk of financial contagion, or the probability that its failure will trigger a chain reaction, leading to the demise of several other banks. A financial system associated with a higher probability of bank failures can be said to be more fragile, and in this case an intervention by the government is conceivably more necessary than in the others. Contagion might follow after the distress of one bank failing due to either solvency or liquidity issues, if there exists a mechanism allowing for the transfer of losses. That is usually the case among banks, since its part of their business to make transactions with each other, e.g., interbank loans. A debtor bank facing problems could simply renege on the payment of a loan, causing a loss to the creditor institution. If the creditor in its turn cannot bear the debtor’s default, the loss will propagate further, characterizing a situation of contagion. Large institutions tend to be the ones with more connections in the market, potentially having a stronger knock-on effect in case they fail. In a netwo rk of banks, those tend to be the first to receive assistance, under the so called too-bi g-to-fail policy, TBTF, by which 2 the government tries to prevent a system’s collapse after a b ank is hit by a shock. The question addressed here is whether a government intervention policy might lead to a morefragile network, wherefragility is viewed as thenumber of bank failures after banks’ assets arehitbyshocks. Thisisdoneby firstconstructinganetwork throughaninteraction process, where banks decide whether or not to be connected, i.e., make an interbank loan to finance investments. Second, with the network in place, shocks are imposed on banks assets, and the number of banks in distress is calculated. The structure of the network turns out to depend on the government’s policy and, therefore, so does fragility, allowing one to study the effects of intervention for that. The model developed is thought of as an economy divided into several regions, with their own idiosyncrasies in terms of investment opportunities and consumers’ preferences. There is a representative bank in each region, responsible for taking deposits and making investments. Investments can bemadein either large or small projects, with larger projects commanding a higher payoff but also demanding an extra level of initial capital, which can be secured only if a loan from another bank is taken. Embedded in the framework is the banks’ maturity mismatch problem, due to the financing of long-term assets with short-term liabilities. Bank are assumed to be short of capital to service depositors, which forces them to sell a fraction of their assets - project or loans - before they are ripe. This happens at a fire-sale cost: there is a penalty applied to any fraction of an asset sold before maturity. The fire-sale c ost is taken to be proportional to the size of the asset, so that large projects are sold at a higher discount - or, in other words, have a lower recovery rate - than small projects and lo ans. Intervention then takes the form of a subsidy offered by the government to banks, so that this fire-sale cost can be alleviated, and such a policy i s assumed to be implemented in a more pronounced way for larger assets. This is how one views a too-big-to-fail type of policy as being present in the framework developed. 3 Thedifferences acrossregions intermsof projects’payoffs andpreferences ofdepositors, combined with the fire-sale cost and government’s intervent ion policy, make some assets to be more profitable than others, according to the region where the bank is located. That will determinethenetwork structureresultingfromaprocessof pairwisemeetings ofbanks, with a link between any two of them representing a loan. The point is that intervention might lead to the creation of links that otherwise would be inexistent, in a way that the structure of the network will depend on the policy chosen by the government. After the formation of the network, the degree of financial fragility is assessed by the number of bank failures caused by having banks’ assets hit by shocks. Upon the shocks, the payoff of projects turn to be only a fraction of what was specified when those projects were undertaken. These are perturbations of the network, in the sense of being probability zero events that banks are not prepared for and, therefore, they immediately cause losses as soon as they take place. Intervention makes the number of links in a network to be at least as high as that of a networkformedundernogovernment. Ceterisparibus,thatmeansthefragilityoftheformer will be at least as high as that of the later, since more banks get exposed to the possibility of contagion. However, even though it is not necessarily the case, a network with a higher number of links might also have a greater overall networth, which means a thicker cushion to absorb shocks and, hence, a smaller number of failures. Therefore, at least theoretically, the effect of government intervention on financial fragility is not straightforward. In order to understand how the impacts of intervention vary with the parameters of the model, simulations are performed for 3 different economies, designed to represent varying stages of financial development. For all of them, the results show that government inter- vention leads to a mucher higher number of bank failures, despite concomitangly increase the overall networh of the banking system, and this effect is stronger the less advanced the economy is. Not only that, leverage might be similar across networks obtained un- 4 der different government policies, even though they present starkingly different degrees of fragility, highlighting the importance of the structure of the system where banks operate and, indirectly, of government intervention for fragility. The paper is structured as follows: following it is presented a simple example that illustrates the main idea of the model; the review of the literature and the contribu- tions/limitations are discussed in subsections 1.1.2 and 1.1.3, respectively; section 2.2 de- tails the model and its primitives; section 1.3 discusses the link formation process and the implications of government intervention for the network structure; section 1.5 gives the balance-sheet characterization of the financial system rep resented by the network, intro- duce shocks and how they can be studied in an input-output ana lysis flavor; section 1.6 proposes differents measures of financial fragility, and a way of ranking banks that happens to be analogous to the PageRank algorithm used by Google to rank websites, detailing also the 3 types of economies designed for the exercise in comparative statics, with qualitative results from the simulations following. 1.1.1 Example Consider two banks, A and B, representing distinct regions of an economy, in a 3-period world, t = 0,1,2. These banks have, at t = 0, the opportunity to invest in local projects paying r A and r B , respectively, at t = 2. The other opportunity available is a 1-period risk-less bond paying 1+ b at t+1, for any $1 invested at t. Without loss of generality, assume that r A > r B > 2. Both projects demand an initial investment of $2, which can be partially supplied by households, who deposit an amount of $1 at t = 0 - the other $1 required to start a project can only be obtained by taking a loan from the other bank, to be repaid at t=1. Householdswithdrawtheirmoneyfrombanksatt=1and,afteraninvestmentismade, banks can obtain the$1 demandedby depositors only by selling a fraction of their projects. 5 This premature selling, at t=1, occurs at a fire-sale price, so that a project which is wort h r i at t = 2 can only be transacted at a price of ρr i at t = 1, for i = A,B. Thus, upon investing in projects, banks need to sacrifice, at t=1, a fraction α i of their investments, so to obtain the amount to service depositors and pay back the loan 1 : α i ρr i = Depositors z}|{ 1 + Loan z}|{ 1+b, ⇔ α i = 2+b ρr i , i=A,B. (1.1) Thus, the profit banks can realize out of projects, at t=2, is Π i =(1−α i )r i ⇔ Π i =r i − 2+b ρ , i=A,B. (1.2) On the other hand, if a bank chooses to invest the $1 from depositors in the risk-free bond, the profit at t =2 is: Π i =[1(1+b)−1](1+b) ⇔ Π i =b(1+b), i=A,B. (1.3) In this way, banks would consider undertaking a project only if that pays better than the risk-free bond: 1 Since banks have the opportunity to invest in a risk-free ass et paying 1+b, they demand an equivalent ammount when lending. 6 r i − 2+b ρ >b(1+b) ⇔ ρ>ρ i := 2+b r i −b(1+b) , i=A,B. (1.4) According to the parameters, the following are the possible scenarios: • Both A and B wants to borrow if ρ>ρ B >ρ A ; • Only A prefers borrowing if ρ B >ρ>ρ A ; and • No bank wants to borrow if ρ B >ρ A >ρ. Considernowthepossibilityofgovernmentintervention, meaningasubsidyofafraction γ of the loss due to the fire-sale cost incurred by banks, 1 −ρ. Upon intervention, thus, the recovery rate of projects increases to ρ+γ(1−ρ), and a project that at t = 1 was worth ρr i is now priced at [ρ+γ(1−ρ)]r i , for i = A,B. The fraction of projects needed to be prematurely sold to service depositors and pay back the loan is now given by: α G i [ρ+γ(1−ρ)]r i = Depositors z}|{ 1 + Loan z}|{ 1+b, ⇔ α G i = 2+b [ρ+γ(1−ρ)]r i , i=A,B. (1.5) Analogously to the previous case, the profit banks can realize out of projects, at t = 2, is: Π G i = 1−α G i r i ⇔ Π G i =r i − 2+b ρ+γ(1−ρ) , i=A,B. (1.6) 7 The condition for banks to be willing to undertake projects rather than the risk-free bond is now: ρ>ρ G i := 1 1−γ 2+b r i −b(1−b) −γ , i=A,B. (1.7) As before, the parameters will determine which of the following scenarios hold: • Both A and B want to invest if ρ>ρ G B >ρ G A ; • Only A considers investing if ρ G B >ρ>ρ G A ; and • No bank wants to invest if ρ G B >ρ G A >ρ. Given the subsidy, the condition for banks to be willing to invest in projects is always easier to be satisfied with intervention than without, i.e., for any γ > 0. Not only that, intervention might lead to bank lending in circumstances where otherwise there would not be any. For instance, if 2+b r A −b(1+b) >ρ> 1 1−γ 2+b r B −b(1−b) −γ (1.8) then it follows that ρ B >ρ A >ρ>ρ G B >ρ G A . (1.9) Therefore, under no intervention, banks would not be willing to borrow, whereas both would like to in case the government is present. One can check that, with r A = 4, r B = 3, b=0, γ =.415 and ρ =.465, condition (1.8) is satisfied. Thus,ifoneassignsallthebargainingpowertobankA, Figure1.1depictsthetwotypes of network that would emerge under different intervention policies: 8 A B Network Structure with Government Intervention A B Network Structure with no Government Intervention Figure 1.1: Network Structure and Government Intervention. Thesetwostructureshavedifferentimplicationsforfinancialfragility. Forinstance,with government intervention, if at t=1 it is anticipated that bank A’s project will be hit by a shock δ A so that its project will be paying r A (1−δ A ) instead, the fraction of the project that needs to be sold to service depositors and pay back the loan is α G A [ρ+γ(1−ρ)]r A (1−δ A )= Depositors z}|{ 1 + Loan z}|{ 1+b, ⇔ α G A = 2+b [ρ+γ(1−ρ)]r i (1−δ A ) , i=A,B. (1.10) If the shock is sufficiently high, i.e., δ A >1− 2+b [ρ+γ(1−ρ)]r A , (1.11) then α G A > 1, hence bank A cannot afford to pay back the loan and service depositors simultaneously, thus it is in default. Assuming that bank B is a second claimant on bank’s A assets - first are depositors - if it still gets some amount fro m the liquidation of assets, i.e., [ρ+γ(1−ρ)]r A (1−δ A )−1>1 ⇔ δ A <1− 2 [ρ+γ(1−ρ)]r A (1.12) 9 then its own default can be avoided, whereas otherwise it fails too. Summarizing: • No bank is in default if δ A ∈ 0,1− 2+b [ρ+γ(1−ρ)]r A ; (1.13) • Bank A is in default but not bank B if δ A ∈ 1− 2+b [ρ+γ(1−ρ)]r A ,1− 2 [ρ+γ(1−ρ)]r A ; (1.14) • Both banks are in default if δ A ∈ 1− 2 [ρ+γ(1−ρ)]r A ,1 . (1.15) Therefore, in case of a sufficiently high shock, the whole network is compromised, since the loss of bank A will cause it to default on the loan taken from bank B, leading to contagion. This possibility is precluded in the network that emerges without government intervention, since in this case both banksare investing in the risk-freebond, which by con- structionmakesthenetworkimmunetoshocks. Inthissense,thus,governmentintervention brings more financial fragility. With the parametrization used previously, any δ A >27.22% leads to the demise of both banks, and no consequences ensue otherwise. Also, measuring thetotal networth of thebankingsystem as thesumof thenetworth of banksA andB, i.e., W =Π A +Π B for the case withoutintervention andW G =Π A +Π B with intervention, one obtains from (1.2), (1.3) and (1.6) that W = 2 and W G = 2.09. Thus, intervention in this example brings more financial fragility, even though it increases the wealth of the banking system. 10 Thefull-fledgedversionofthemodelhasmanybanks,withtwo investmentopportunities available - large and small projects - on top of the possibili ty of lending. Also, banks are equipped with different levels of capital and varying amounts of households withdrawing their deposits before projects are ripe, causing a maturity mismatch problem. As in the example, government intervention takes theformof asubsidythatallows banks toalleviate theirfire-salecosts,andmightturnprofitableinvestmento pportunitiesthatotherwisewould notbe. Thiswillleadtodifferentnetworkstructures,withdistinctlevelsoffinancialfragility associated to them. 1.1.2 Related Literature Since the main theme of the paper is how government intervention can lead to financial fragility, the related literature is traced to that on financial crisis, contagion, government intervention and networks. For each of these, the papers closer in spiritto the present work are highlighted. In the financial crisis literature, the main reference is the classic Diamond and Dybvig (1983) work on bank runs. It is from their paper the idea of having depositors with con- sumption needs arising stochastically. Differently from Diamond and Dybvig, though, the framework presented has multiple banks and studies how distress in one institution spreads to others, the mechanism of contagion beingthe links across banks arising dueto interbank lending. Another important reference is Shleifer and Vishny (1992), providing the ratio- nale for having a fire-sale cost adventing from the early sell ing of projects, which motivates government intervention in the model. Network theory has been increasingly used to study different issues in economics and finance. Allen and Babus (2009) provide an account with specific applications to finance and Schweitzer, Fagiolo, Sornette, Vega-Redondo, Vespign ani, and White (2009) point to new directions of research. One approach is to study how systemic risk is associated with 11 different types os networks, i.e., the susceptibility to contagion. Some papers along these lines are: Rochet and Tirole (1996), where banks have a role in monitoring each other and whose closure decisions after being hit by a shock are interlinked; Kiyotaki and Moore (1997), withachainoffirmsborrowingandlendinggivingrisetosystemicrisk,incasesome ofthembecometemporarilyilliquidandcauseotherstogetintofinancialdifficultiesaswell; Allen and Gale (2000b), which shows that similarly efficient networks have varying degrees of robustness to shocks; Freixas, Parigi, and Rochet (2000), with credit lines across banks in different regions being the channel for contagion; Eisenberg and Noe (2001), developing a model of a clearing system and providing a measure of systemic risk based on the number of “waves” of default necessary to cause the bankruptcy of a b ank; Lagunoff and Schreft (2001), where diversification leads agents to have their portfolios linked, with these connec- tionsbeingsubjecttobebrokenasaresultofassetreallocation following shocks, eventually leading to additional disruptions and a crisis; Cifuentes, Ferrucci, and Shin (2005), which studies how contagion might be processed not only through direct balance sheet exposure among banks but also via asset prices, addressing also the consequences that might advent from prudential regulation; Nier, Yang, Yorulmazer, and Alentorn (2007), first modelling banking systems as random graphs and then simulating changes in the underlying param- eters, in order to assess the resilience to shocks; Brusco and Castiglionesi (2007), with liquidity coinsurance potentially bringing down a bank in case it is paired with a not well capitalized institution that engages in excessive risk taking; more recently, Caballero and Simsek (2011), with a model showing how uncertainty about the network structure, coined complexity, exacerbates shocks and makes banks to behave more precautiously, leading to fire-sales and crises; Zawadowski (2011), who uses a network framework to model bilateral over-the-countercontracts, showingthatbanksunderinsu reagainstcounterpartyriskbynot incorporatingthenetwork externalities they imposeonthirdparties oncetheyfail; Haldane and May (2011), drawing on an analogy of the banking system to ecologic food webs and 12 networks where infectious diseases spread; and Duffie (2011), discussing a network-based approach to monitor systemic risk. What these papers do not provide is an explicit mechanism that leads to the forma- tion of financial networks. Some that do are: Leitner (2005), with banks forming links that allow transfers of endowments to be made, which in turn prevents the failure of less wealthy members that could cause the demise of the entire network; Babus (2009), using a game-theoric approach to endogenously derive a network th at can provide insurance to the possibility of contagion; Castiglionesi and Navarro (2011), where banks decide whether to join a network that makes it possible for them to coinsure each other against liquidity shocks, throughthegrantingofcreditlinesthatinturnresultinthembeingconnected; and Cohen-Cole, Patacchini, and Zenou (2012), which model the n etwork formation process as a Cournot competition in the lending market, showing that banks prefer to be linked with others that are more rather than less connected. Related to government intervention, some papers that study the effects on the behavior of private institutions from government’s policy in periods of distress are: Huang and Xu (1999), whichmodelthe1997EastAsiafinancialcrisisasaresultofsoftbudgetconstraints; SchneiderandTornell(2004), which, inatwo-sector econom y, showthatacontract enforce- abilityprobleminducesbailoutguaranteestonontradables,propellingthatsectortoinitiate crises; Gorton and Huang (2004), wheregovernment bailouts can bean efficient mechanism for the recapitalizion of banks in times their assets are hit by a negative shock, by means of providing liquidity in the secondary market for projects and hence guaranteeing they can fetch a price better than what a fire-sale would entail; Corse tti, Guimarães, and Roubini (2006) and Morris and Shin (2006), studying models of catalytic finance in which, rather than moral hazard, support from institutions of the like of the IMF ends up being pivotal for countries to engage in costly but necessary reforms; Acharya and Yorulmazer (2007), analysingtheimplications of intervention policies thataredesignedtobeimplemented only 13 in systemic crises - those that affect a significant portion of thebankingindustry- resulting in a too-many-to-fail type of guarantee, which in turn incre ases the likelihood of distress in the bankingsystem; Ennis and Keister (2009), whereex post efficient interventions to bank runs might generate self-fulfilling crises episodes that de stabilize the banking system; Farhi andTirole(2010) andDiamond andRajan (2011), bothwithmodels analysingintervention through changes in the interest rate, Farhi and Tirole showing that a policy whereby the government respondstocrisesdecreasing theinterest rateleads tomorematurity mismatch in the economy and, thus, exposure to liquidity shocks, whereas Diamond and Rajan argue that such a policy turns out to be better than the alternatives, in particular a bailout of a specific institution. The sources of distress analysed are payoff shocks, i.e., a decrease in the payment of projects. Thepaperabstractsfromprovidingareasonforwhysuchshockstakeplace. Some papers in that regard, to cite a few, are: related to bubbles, Allen and Gale (2000a) and Abreu and Brunnermeier (2003); the leverage cycle theory, Geanakoplos (1997), Geanako- plos (2003) and Geanakoplos (2010); liquidity spirals, Brunnermeier and Pedersen (2009); imperfect information, Morris and Shin (1998) and He and Xiong (2011); flight to quality, global imbalances and sudden stops, Caballero and Krishnamurthy (2008), Mendoza and Quadrini (2010) and Mendoza (2010), respectively. 1.1.3 Contributions of the Paper The main contribution of the paper is to provide a framework based on networks to ad- dress the question of whether government intervention can lead to financial fragility. The framework developed is essentially an attempt to combine the Diamond and Dybvig (1983) model of bank runs with the Allen and Gale (2000b) model of financial contagion, with two additional components: government intervention and network formation. In so doing, it provides a mechanism where government’s policy is crucial in the banks’ 14 decision to become or not connected, i.e., create links. To the best of one’s knowledge, this is the first paper that models the formation of a network of banks with an explicit role for the government. Obviously, as pointed out in the previous section, it is neither the first paper to model the formation of financial networks nor government intervention, it only combines these two aspects. With the government being pivotal in the process of network formation, the framework provided allows one to study the effects of particular policies, in particular a too-big-to-fail type of intervention, whereby large enterprises (projects in the model) enjoy a higher level of assistance than small ones. The financial crisis of 2007-2 009 produced a lot of debate about this type of intervention and the potential consequences arising from that. Several papers discuss how government bailouts can lead to moral hazard, but the effects on the incentives for banks being connected, using a network framework as it is proposed, seems to be novel. The model proposed is very tractable and allows one to perform different kinds of sim- ulations, in order to assess the implications of alternative government policies. Not only that, one can also use it as a stress test tool, to measure how the network is vulnerable to the distress of particular financial institutions. All in all, and as the simulations show, the point is that knowledge of the network structure of the financial system - as far as financial fragility is concerned - is as valuable of an information as a re more standard measures, such as the leverage ratio. Being the government crucial for network formation process, so crucial it is its policy in determining the degree of financial fragility. However tractable the model is, it is also a very stylized way of seeing reality. For instance, the network formation process implicitly assumes that banks are myopic, in the sense that they make decisions without considering other banks they meet in the future. Also, one abstracts from the budget constraint of the government, which is viewed simply as an institution with deep pockets that is there offering a subsidy to banks, to offset 15 some of their costs. The plausibility of the government behaving in such a way is obviously questionable, andinfullfledgedmodelitwouldneedtobereconsidered. Anotherimportant aspectisthat,byhavingnouncertainty,themodelabstractsfromthedefaultriskthatbanks calculate when they decide whether or not to lend to other institutions. The huge volume of literature on default risk highlights how important such an aspect is but, for the sake of tractability, this aspect is not considered in the framework proposed. 1.2 Model Consider a 1-good ($), 3-period economy, t = 0,1,2, divided in an even number N of regions, N = { 1,...,N} . Every region i ∈ N has a representative bank B i ∈ B, with B ={ B 1 ,...,B N } representing the set of banks, each with a secured endowment (equity) of e i >0 per transaction they engage in, as it will be explained later. Every region i hasN−1 continuums of depositors,D i = n D i 1 ,...,D i N−1 o , each of them of unit mass. Depositors are endowed with $1 and have Diamond-Dybvig preferences, i.e., they face uncertainty regarding the time of consumption, which is formalized by having the following utility function: U i (c 1 ,c 2 )=        c 1 , with probability ω i , c 2 , with probability 1−ω i . (1.16) For anycontinuumD i j inregion i, with probabilityω i adepositorwill consumeatt=1, denoted by c 1 and denominated henceforth early depositor, whereas with probability 1−ω i she will consume at t = 2, denoted by c 2 and denominated late depositor. Uncertainty is resolved at t = 1 and the probability of consuming at t = 1 or t = 2 varies across regions. Figure 1.2 illustrates the continuums of depositors for a region i∈N. Any representative bank has available an infinite supply of two types of long-term in- 16 1−ωi ωi 1−ωi ωi 1−ωi ωi 1−ωi ωi D i 1 D i 2 D i N−1 D i 3 Region i Figure 1.2: Region i and its continuums of depositors. vestment opportunities, namely a large and a small project. At t = 2, the large project pays r ∗ i whereas the small yields r i , the former for an investment of $2 and the later of $1, at t = 0. By assumption, r ∗ i > r i , for any i ∈ N. The cash flows of projects available to bank B i ∈B are represented in Figure 1.3. t=0 t=1 t=2 -2 r ∗ i Large Project t=0 t=1 t=2 -1 r i Small Project Figure 1.3: Cash flows of region i’s projects. Projects are available only to the representative bank of the respective region, i.e., a bank i ∈ N cannot invest in projects other than the ones in its own region - cross-region investment is ruled out 2 . Projects can be partially sold before maturity, i.e., at t = 1, at a fire-sale price, as it will be explained in the sequence. Ban ks are also allowed to borrow (long-term) from other banks and they also have available at t=0 a short-term asset that pays zero interest rate. Depositors do not have access to either long-term projects or short- 2 One way of thinking about this is that projects require an expertise that only local banks have. 17 term assets, and are forced to deposit their endowments in the local bank, by means of a deposit contract that allows withdrawals at will. This imples that banks can borrow from depositors at a zero interest rate. 1.2.1 Banks Interaction Process and Arrival of Depositors Banks interact with each other in order to take advantage of differences across regions in the payoff offered by small and large projects, since cross-re gion investment is ruled out. The interaction protocol of banks is assumed to be such that, at t = 0, a specific number of rounds of interaction takes place, with banks meeting pairwise. The number of rounds is such that, at the end of the interaction process, banks will have met each other once, and only once. Given an even number N of banks, at t = 0 there will be N −1 rounds of interaction. For example, with four banks, N = 4, the rounds of interaction will be as in Figure 1.4: B 1 B 1 B 1 B 3 B 2 B 2 B 4 B 4 B 3 B 2 B 3 B 4 Round1: Round2: Round3: ; ; ; Figure 1.4: Interaction process of banks for N =4. Synchronized with the interaction process of banks is the arrival process of depositors. Depositors show up sequentially at their local banks, in a time fashion matching the way that banks meet. In every round of interaction, one of the N−1 continuums of depositors in each region arrives at the local bank and, given that the interaction process of banks is composed of N −1 rounds, in the end of it all of the continuum of depositors will have made deposits at their respective banks. 18 1.2.2 Network Structure At every round of interaction, a bank (i) receive depositors and a specific amount of en- dowment, and (ii) meet another bank, when it decides on making an investment in either a loan or a project. An important assumption is that banks need to decide during the round of interaction how to allocate the $1 received from depositors. This implies banks behaving in a myopic way, since they cannot keep the money received for more profitable transactions that might come in the future. At each round of interaction, banks have to decide on whether to: (i) Invest the $1 received in a small project; (ii) Borrow an extra $1 and invest the total in a large project; (iii) Lend $1 to the other bank. Figure 1.5 illustrates, for a particular round of interaction, the possibilities arising from a meeting between banks i and j. The blue dashed line represents an investment in a small project, the green in a large project in region j - by means of a loan agreement between bank i (lender) and bank j (borrower) - whereas the red an investment in a large project in region i, with the roles of bank i and j switched. Theconditionforaloanagreementtotakeplaceisthepaymentbytheborrower(interest plusprincipal)tobeatleastaslargeastheopportunitycostofthelender. Theopportunity cost of the lender arises from the fact that, by disposing of $1, the possibility of investing in a small project is foregone. Therefore, in any loan agreement, the borrower must pay to the lender at least the payoff the later would get by investing in a small project. Upon the meeting of banks i and j, the double-headed arrows in Figure 1.4 turn into one of the following: (i) B i −→ B j : B i lends to B j ; 19 Bank i Bank j Large Projectj Large Projecti Small Project i Small Project j r ∗ i r ∗ j r j r i r i r j 1−ωi ωi ωi 1−ωi ωi 1−ωi ome3 1−ωj ωj ωj 1−ωj ωj 1−ωj Region i Region j 1−ωi ωi D i n 1−ωj ωj $1 $1 $1 $1 $1 $1 $1 $1 $2 $2 D j n Figure 1.5: Portfolio decision of banks at a particular round of interaction. (ii) B i ←− B j : B i borrows from B j ; (iii) B i ··· B j : No loan agreement between B i and B j . Therefore, with N banks, after N − 1 rounds of interaction many types of network structures might emerge. As an example, Figure 1.6 illustrates possible strucutures when N =4. 1.2.3 Maturity Mismatch To finance an investment in either a small project or a loan, banks need deposits and, in case of large projects, to borrow money from other banks. Since assets payoff only in the long-term whereas a fraction of deposits is withdrawn in the short, banks are exposed to 20 B 1 B 2 B 4 B 3 B 1 B 2 B 4 B 3 B 1 B 2 B 4 B 3 B 1 B 2 B 4 B 3 Figure 1.6: Examples of networks at the end of the interaction process (N =4). the problem of maturity mismatch, i.e., the use of short-ter m funds to finance long-term assets. Any bank i∈ N has available and endowment of e i , for every transaction they engage in, either an investment in a project or a loan. However, banks are assumed to be wealth constrained, meaning that e i is not sufficient to cover withdrawals by early depositors, ω i . Therefore, for any bank i∈I, it is assumed that e i <ω i . Being wealth constrained, early withdrawals can be met only by having banks prema- turely selling a fraction of their investments. This early selling of assets occurs at a fire-sale price. How costly it is to sell assets is assumed to depend on the size of the investment made, in the following way: (i) Large projects have a recovery rate of ρ ∗ : one unit of investment in a large project paying r ∗ i at t=2 is priced at ρ ∗ r ∗ i at t=1, with 0<ρ ∗ <1; (ii) Small projects have a recovery rate of ρ: one unit of investment in a small project paying r i at t=2 is priced at ρr i at t=1, with 0<ρ<1. Since loans are always the size of an investment in a small project, $1, the fire-sale 21 cost associated to them is taken to be the same as the one for small projects, ρ. Another important assumption is that the fire-sale cost of large proj ects is higher than that of small ones (and loans), i.e., 0<ρ ∗ <ρ<1. (1.17) Government intervention, as discussed next, is a way of alleviating the costs imposed on banks due to this premature selling of assets. 1.2.4 Government Intervention The premature selling of assets is costly and, depending on how severe this cost is, banks might prefer to make small rather than large investments, since the former type of projects are less costly to be transacted than the later ones. For reasons abstracted from, it might bein the interest of the government to reduce the fire-sale cost of banks. Intervention takes the form of a subs idy offered by the government to thebanks, and is thoughtof as if therewas a secondary market allowing for thepurchase of distressed assets and, by actively participating on that, it would enable banks to fetch a better price for their investments. It is assumed that intervention changes the recovery rate of projects as in the following: (i) Large projects: with intervention, one unit of investment in a project paying r ∗ i at t=2 is priced at [ρ ∗ +γ ∗ (1−ρ ∗ )]r ∗ i at t =1; (ii) Small projects: with intervention, one unit of investment in a project paying r i at t=2 is priced at [ρ+γ(1−ρ)]r i at t=1. Thus, with γ and γ ∗ defined as the government intervention parameters for small and large projects, respectively, under no intervention, i.e., γ ∗ = γ = 0, the original recovery 22 rates apply, ρ for small projects and ρ ∗ for large ones. On the other hand, with full govern- ment participation, γ ∗ = γ = 1, there is no fire-sale cost to be incurred when projects are negotiated. In order to capture the effects of what is thought as a too-big- to-fail type of policy, it is assumed that large projects command more support from the government than small ones, i.e., γ ∗ >γ. Despite such a policy, however, large investments are still assumed to be more costly to be sold than small ones, i.e., ρ+γ(1−ρ)>ρ ∗ +γ ∗ (1−ρ ∗ ). (1.18) 1.2.5 Timeline of Events With the ingredients of the model in hand, the timeline of events is the following: • t=0: 1. Banks meet pairwise, giving rise to a network structure after the interaction process. At each round of meetings banks decide: (i) Whether or not to form a link (make a loan or borrow); (ii) How much to invest in the short-term asset; (iii) How much of the long-term asset (project or loan) to sel l in order to meet early withdrawals. • t=1: 1. Banks execute the selling strategy; 2. Together with the investment in the short-term asset, pro ceeds are used to pay early depositors. 23 • t=2: 1. Payoffs from long-term assets (projects and loans) are rea lized; 2. Banks pay late depositors and clear positions with other banks, consuming the remainings as profits. The next section details the link formation process of banks. 1.3 Link Formation After all the rounds of interaction, the network structure that arises is the result of banks’ borrowing and lending decisions made at each of the pairwise meetings they participate. Every time a loan is made a link in the network is formed. For that, consider any two arbitrary banks, i and j. Without loss of generality, one focus on each of the three possible choices of bank i when meeting bank j. 1.3.1 Investment in a Small Project If banki is to invest in a small project, the budget constraints to besatisfied in each period are: 1+b i ≤1+e i (BC at t=0) b i +α i r r i [ρ+γ(1−ρ)]≥ω i (BC at t=1) 1−α i r r i =(1−ω i )+e i +π i (BC at t=2) (1.19) where: • π i : profit of bank i with an investment in a small project; 24 • b i : investment in the short-term asset (bond that pays no inter est); • α i r : fraction of the small project to be liquidated at t=1. The budget constraint at t = 0 expresses that total expenses, i.e., investment in the small project, $1, plus investment in the short-term asset, b i , cannot exhaust the amount of total resources available, namely deposits, $1, and equity disbursement, e i . At t=1, the revenue fromthe short-term asset, b i , plusthe proceedsfrom theliquidation of a fraction of the small project, α i r r i [ρ+γ(1−ρ)], should suffice to service early depositors, ω i . Finally, at t = 2, the fraction not liquidated of the small project, 1−α i r r i , must allow the bank to meet the demand from late depositors, 1−ω i , plus the amount owed to equity holders, e i . What is left from the payoff of the small project after paying late depositors and equity holders constitutes the profit of the bank, π i . Since bank i does not want to (i) waste resources and (ii) sell a higher fraction of the small project than what is necessary, the budget constraints at t = 0 and t = 1 will be binding, allowing one to solve for b i and α i r : b i = e i , (1.20) α i r = ω i −e i r i [ρ+γ(1−ρ)] . (1.21) Substituting into the expression for bank i’s profit, π i becomes: π i =r i − (1−ω i )+e i + ω i −e i [ρ+γ(1−ρ)] ⇔ π i =r i −r i , (1.22) where 25 r i :=(1−ω i )+e i + ω i −e i [ρ+γ(1−ρ)] . (1.23) Obviously, bank i would be willing to accept deposits that it could channel to a small project as long as π i ≥0, i.e., r i ≥r i , which is an assumption maintained for any i∈N. 1.3.2 Investment in a Large Project If bank i is to invest in a large project, a loan agreement has to be established so that $1 is borrowed from bank j. The budget constraints are then modified in the following way: 2+b i ≤2+e i (BC at t=0) b i +α i r ∗r ∗ i [ρ ∗ +γ ∗ (1−ρ ∗ )]≥ω i (BC at t=1) 1−α i r ∗ r ∗ i =(1−ω i )+e i +y ij +π ∗ ij (BC at t=2) y ij ≥r j (IR of the Lender) (1.24) Differently from an investment in a small project, in the budget constraint at t = 0 there is now one extra $1 coming from the loan taken and, as a result, at t= 2 there is an extra expense representing the loan 3 that bank i has to repay to bank j, an yield denoted by y ij . The profit of the bank in a large project in turn depends from whom the lender is and, accordingly, is denoted by π ∗ ij . An additional constraint in bank i’s problem is the individual rationality constraint of bank j, the lender, labeled IR. By extending a loan, the lender incurs an opportunity cost equal to the payoff it could have gotten by investing in a small project. Therefore, bank j is willing to participate in a loan agreement as long as y ij ≥r j . Without loss of generality, 3 Principal plus interest. 26 the borrower is assumed to have all the bargaining power and, as such, offers the minimum interest on the loan, at which the lender is indifferent between lending or not 4 , resulting in y ij =r j . Analogously to an investment in a small project, the budget constraints at t = 0 and t=1 bind, for there is no reason why bank i would either waste resources neither over sell its project. Adding to that the fact that bank i pays the minimum interest to the lender, the following results: y ij = r j , (1.25) b i = e i , (1.26) α i r ∗ = ω i −e i r ∗ i [ρ ∗ +γ ∗ (1−ρ ∗ )] . (1.27) Bank i’s profit with a large project when borrowing from bank j, π ∗ ij , is then: π ∗ ij =r ∗ i − (1−ω i )+e i +r j + ω i −e i [ρ ∗ +γ ∗ (1−ρ ∗ )] . 1.3.3 Investment in Loans From theassumptionsthat(i)thediscountparameter for loans isthesameappliedtosmall projects and (ii) the payoff on a loan is the same as the one in a small project, loans and small projects are perfect substitutes. The same analysis used for small projects, therefore, applies to the case of loans: for bank i, the investment in the short-term asset and the fraction of the long-term investment to be sold before matur ity - loans instead of small projects - will be the same, and so will be its profits. For prac tical purposes, therefore, 4 One way of breaking the indifference point towards bank j extending a loan would be to impose a cost due to asymetric information with investments in projects. Since, presumably, market forces lead banks to be more scrutinized than projects, a loan to another bank would be preferable to an equivalent investment in a small project, ceteris paribus. 27 investments in loans and small projects are indistinguishable. 1.4 Banks’ Portfolio Allocation Comparing the profits that could be obtained in each of their investment opportunities, banks i and j will decide whether to (i) stay in autarky, i.e., invest separately in their respective small projects, or (ii) make an interbank loan, allowing the borrower to invest in a large projectand thecreditor in aloan. Thepossibleoutcomes after the pairwisemeeting of any two banks i and j are summarized in the following: 1. Bank i wants to borrow from bank j, but not vice versa: bank i is better-off investing in a large project, π ∗ ij > π i , whereas the opposite is true bank j, π j > π ∗ ji . From expressions (1.22) and (1.28), that is the case if the following holds: r ∗ i −(r i +r j ) ω i −e i > [ρ+γ(1−ρ)]−[ρ ∗ +γ ∗ (1−ρ ∗ )] [ρ+γ(1−ρ)][ρ ∗ +γ ∗ (1−ρ ∗ )] > r ∗ j −(r j +r i ) ω j −e j , (1.28) where the first inequality represents the fact that π ∗ ij > π i , whereas the second that π j >π ∗ ji . 2. Bank j wants to borrow from bank i, but not vice versa: Analogous to the previous condition, but now with bank j being the one interested in borrowing, π ∗ ji >π j , and bank i in lending, π i >π ∗ ij : r ∗ j −(r j +r i ) ω j −e j > [ρ+γ(1−ρ)]−[ρ ∗ +γ ∗ (1−ρ ∗ )] [ρ+γ(1−ρ)][ρ ∗ +γ ∗ (1−ρ ∗ )] > r ∗ i −(r i +r j ) ω i −e i , (1.29) where the first inequality represents the fact that π ∗ ji > π j , whereas the second that π i >π ∗ ij . 28 3. Both banks i and j want to borrow: In this case, it holds that π ∗ ij > π i and π ∗ ji >π j , translated into: min ( r ∗ i −(r i +r j ) ω i −e i , r ∗ j −(r j +r i ) ω j −e j ) > [ρ+γ(1−ρ)]−[ρ ∗ +γ ∗ (1−ρ ∗ )] [ρ+γ(1−ρ)][ρ ∗ +γ ∗ (1−ρ ∗ )] . (1.30) In this scenario, the tie is broken by favouring the bank which is to make the larger profit from borrowing: from (1.28), if π ∗ ij >π ∗ ji , i.e. [r ∗ i −(1−ω i )−e i −r j ]− h r ∗ j −(1−ω j )−e j −r i i > (ω i −e i )−(ω j −e j ) ρ ∗ +γ ∗ (1−ρ ∗ ) , (1.31) then bank i ends up borrowing from bank j; otherwise, i.e., if π ∗ ji >π ∗ ij , h r ∗ j −(1−ω j )−e j −r i i −[r ∗ i −(1−ω i )−e i −r j ]> (ω j −e j )−(ω i −e i ) ρ ∗ +γ ∗ (1−ρ ∗ ) , (1.32) then it is bank j who borrows from bank i. 4. Neither bank i nor bank j wants to borrow: in this scenario, it follows that both banksare better-off investing in their respective smal l projects, with π i >π ∗ ij for bank i and π j >π ∗ ji for bank j holding, i.e. [ρ+γ(1−ρ)]−[ρ ∗ +γ ∗ (1−ρ ∗ )] [ρ+γ(1−ρ)][ρ ∗ +γ ∗ (1−ρ ∗ )] >max ( r ∗ i −(r i +r j ) ω i −e i , r ∗ j −(r j +r i ) ω j −e j ) . (1.33) In this scenario, therefore, banks remain in autarky. 29 1.4.1 Government Intervention and Network Structure Theconditionsfortheformationoflinksacrossbanksgivenpreviouslyholdforanyarbitrary set of parameters. In particular, the conditions are true for γ ∗ = γ = 0, the case where there is no government intervention. One is interested in seeing how adding government intervention, γ ∗ ≥ γ > 0, affects the structure of a network, in particular regarding the number of links, i.e., interbank loans. Proposition 1.1. For any set of parameters, the network that emerges under government intervention has at least the same number of interbank loans (links) as the network that emerges under no government intervention. Proof: See Appendix. Proposition 1.1 establishes that government intervention does not brake links across banks that would exist otherwise, and eventually it leads banks to make new interbank loans. In other words, intervention makes the network of banks to be more connected. One might wonder if it is possible after intervention that an otherwise borrower would become a lender, considering an arbitrary pair of banks. According to Proposition 1.2, the answer to this question is yes. Proposition 1.2. Consider a network of banks formed without government intervention. If two arbitrary banks engage in a loan agreement, the identity of the borrower and the lender might change if one considers the network that would prevail with the participation of the government. Proof: See Appendix. The reason for the above result is the following: under no intervention, suppose that bank i is better-off borrowing and bank j lending. That does not mean that bank i’s profit is higher than bank j’s, it is just that bank i’s profit investing in a large project is higher 30 than what it could get with a small one, and vice versa for bank j. With government intervention, the scenario will remain the same for bank i, but it might change for bank j and, in this case, both banks would bewilling to borrow to finance an investment in a large project. In this case, the tie-breaking rule might favor ban k j, leading to a change in the identity of the borrower and the lender for this particular interbank loan. Government intervention, therefore, has a pivotal role in determining the shape of the network structure emerging from the interaction among banks. As a corollary of Lemma A.1, it also follows that under the so called too-big-to-fai l policy, TBTF, γ ∗ > γ > 0, the incentives for the creation of links are even stronger, vis- a-vis what would result in case the government subsidy for large and small projects was to be the same, γ ∗ =γ >0. Corollary1.1. Under aTBTFpolicy, γ ∗ >γ >0, the incentivescreated bythe government for the banks to get connected are even stronger. Proof: See Appendix. Government’s policy will determine the structure of the network that emerges, given the other parameters of the model. This will lead to a matrix representation of banks connectedness, and with that the whole financial system, i.e., the balance-sheet of banks, can be characterized. With that in hand, one can study the fragility of the banking system by imposing shocks on banks assets and seeing how they propagate. This is what is done in turn. 1.5 Characterization of the Financial System and Shocks Followingthepairwisemeetingsofbanksineachroundofinteraction, thenetworkstructure - the financial system - can be describe by the following matri x: 31 X =             0 χ 12 ··· χ 1N χ 21 0 ··· χ 2N . . . . . . . . . . . . χ N1 χ N2 ··· 0             , (1.34) where χ ij is an indicator function such that χ ij = 1 if i lends to j and 0 otherwise. From now on, one assumes that the quantities being discussed are for an arbitrary bank i, with i∈ N. The row-sum gives the number of debtors, i.e., the number of loans made, n i L = P j∈N χ ij , whereas the column-sum gives the number of creditors which , equivalently, is the number of large projects undertaken, n i r ∗ = P j∈N χ ji . Since in every round of interaction banks invest in an asset, and there are N −1 of those rounds, the number of small projects can be obtained by n i r =(N−1)−n i L −n i r ∗. Together with the fire-sale parameters for small and large pr ojects, ρ and ρ ∗ , and the respective parameters for government intervention, γ and γ ∗ , the primitives of the model are: r ∗ =             r ∗ 1 r ∗ 2 . . . r ∗ N             r =             r 1 r 2 . . . r N             ω =             ω 1 ω 2 . . . ω N             e=             e 1 e 2 . . . e N             . (1.35) Recall that banks cannot serve early depositors only with equity disbursements, i.e, ω i > e i , for any i ∈ N. This leads to a premature liquidation of a fraction α i r of an investment in either a loan or a small project, as in (1.21), and a fraction α i r ∗ in case the investmentisinalargeproject,asin(1.27). Att=2, therefore, thereisa i L =r i 1−α i r n i L worth in loans, a i r ∗ = r ∗ i 1−α i r ∗ n i r ∗ in large projects and a i r = r i 1−α i r n i r in small projects. Assets are then written as a i =a i L +a i r ∗ +a i r . 32 Liabilities are composed by the amount owed to late depositors, loans taken from other banks, i.e., debt, and networth, which is equity plus profits. There is $1 collected from depositors in all the N −1 rounds of interaction, and the fraction of these deposits that remains to be claimed at t = 2 is 1− ω i . Therefore, the amount owed to depositors is l i ω =(N−1)(1−ω i ). Debt is owed to other banks due to loans taken in order to invest in large projects and, therefore, is written as l i d = P j∈N χ ji r j . As for the networth, equity is given by l i e =(N −1)e i , since thereis an equity disburse- ment in all the rounds of interaction. Profits are obtained from investments, composed of loans and projects. Loans and small projects yield the same profit, π i , and since there are n i L loans and n i r small projects, profits from these two assets is n i L +n i r π i . From large projects, the profit is π ∗ ij when the loan is taken from bank j and, therefore, profits gener- ated from going large are P j∈N χ ji π ∗ ij . Profits are then l i π = n i L +n i r π i + P j∈N χ ji π ∗ ij . Networth is writen as W i = l i e + l i π and, therefore, liabilities in the balance-sheet are l i =l i ω +l i d +W i . Inthisway, foranynetworkformedwithorwithouttheparticipationofthegovernment, one can obtain the level of assets, investments in small and large projects, debt of banks, networth, leverage andothermeasuresrelated tothebalance-sheet ofbanks. Aninteresting feature of the participation of the government in the network formation process is that the total networth of thefinancial system doesnotnecessarily increase. Thisis acorollary from Propositions 1.1 and 1.2: Corollary 1.2. Let π G := min n π G 1 ,...,π G N o and π := max{ π 1 ,...,π N } denote the mini- mum payoff of a small project among all the banks, with and without intervention, respec- tively. If π G >π, then government intervention leads to an increase in the networth of the network of banks, P i∈N W i . Otherwise, that is not necessarily the case. Proof: See Appendix. 33 Corollary 1.2 is important for the study of the effects of government intervention on the capacity of networks to absorb shocks - to be defined next - giv en that the only cushion banks have to deal with losses is their networth. The shocks to be introduced affect the payoffs that banks are entitled to receive from their investments in projects, either large or small. For instance, if bank i has made an investment in a large project with a payoff of r ∗ i , a shock of δ ∗ i implies that it receives r ∗ i (1−δ ∗ i ) instead. If the investment was in a small project, the payoff is r i (1−δ i ) rather than r i , considering a shock of δ i . The vectors of shocks in the payoffs of small and large projects are given, respectively, by: δ =             δ 1 δ 2 . . . δ N             δ ∗ =             δ ∗ 1 δ ∗ 2 . . . δ ∗ N             . (1.36) Upon being hit by shocks, banks face a loss in the asset side of their balance-sheets, whereas their liabilities remain the same. The loss is written as Δ i = Δ i L + Δ i r ∗ + Δ i r , explained in the sequence. The loss with small projects is Δ i r = a i r δ i , whereas it is Δ i r ∗ = a i r ∗δ ∗ i with large projects. The loss with loans, Δ i L , will depend on the bankruptcy status of bank i’s debtors and, therefore, it is endogenously determined. A bank is bankrupt if it cannot fulfill entirely its obligations with debtors and households, i.e., if the losses incurred are greater than the networth, Δ i >W i . For instance, assume that bank i has lent to bank j, i.e., χ ij = 1, and bank j is bankrupt. The losses spread by bank j are given by Δ j −W j > 0, first evenly distributed among debtors, which are owed the amount l j d , and, if greater than that, spread also to 34 households. Therefore, for every unit borrowed, bank j pays only 5 Δ j −W j /l j d . Upon that, the indirect loss to bank i, which expected to receive a payment of r i for the fraction 1−α i r of the loan still in its balance-sheet, is r i 1−α i r Δ j −W j /l j d . Therefore, Δ i L =r i 1−α i r X j∈N χ ij Δ j −W j + l j d (1.37) represents theindirect losses suffered by bankiupontheeventual defaultof its debtors, where() + denotesthepositivepart. Thesystemofequationsthatendogenouslydetermines e Δ i := Δ i −W i + - bank’s loss in excess of networth - is: e Δ 1 =   r 1 1−α 1 r X j∈N χ 1j e Δ j l j d +Δ 1 r ∗ +Δ 1 r −W 1   + , (1.38) e Δ 2 =   r 2 1−α 2 r X j∈N χ 2j e Δ j l j d +Δ 2 r ∗ +Δ 2 r −W 2   + , (1.39) . . . e Δ N =   r N 1−α N r X j∈N χ Nj e Δ j l j d +Δ N r ∗ +Δ N r −W N   + . (1.40) In matrix form, this system is written as: e Δ= e X e Δ+Δ r ∗ +Δ r −W + , (1.41) where e X e Δ represents the vector of indirect losses, Δ r ∗ of losses in large projects, Δ r in small and W the networth, the dimensions of these vectors being N × 1. The matrix e X is given by: 5 One assumes that depositors are protected by deposit insurance so the focus is on the case where Δ j −W j ≤l j d . 35 e X =             0 r 1(1−α 1 r )χ 12 l 2 d ··· r 1(1−α 1 r )χ 1N l N d r 2(1−α 2 r )χ 21 l 1 d 0 ··· r 2(1−α 2 r )χ 2N l N d . . . . . . . . . . . . r N(1−α N r )χ N1 l 1 d r N(1−α N r )χ N2 l 2 d ··· 0             , (1.42) which can be equivalently written as e X =diag             r 1 1−α 1 r r 2 1−α 2 r . . . r N 1−α N r                         0 χ 12 ··· χ 1N χ 21 0 ··· χ 2N . . . . . . . . . . . . χ N1 χ N2 ··· 0             diag             1 l 1 d 1 l 2 d . . . 1 l N d             . (1.43) The matrix e X is simply a weighted version of X, the matrix representing the network structure, row-weights being the expected payoff on loans ma de that are still on banks’ balance-sheet, r i 1−α i r , and column-weights being each bank’s per unit amount owed relative to total debt, 1/ P k∈N χ ki r k . From (1.37), an element i,j of e X represents the per unit loss that bank j imposes on bank i, conditional on i having lent to j, i.e., χ ij =1. One proceeds with a thought experiment and assume that banks have zero networth, i.e., W i =0, for i∈N. Expression (1.41) is then written as: e Δ= e X e Δ+Δ r ∗ +Δ r , (1.44) and, upon some conditions to be discussed being satisfied, e Δ= I− e X −1 (Δ r ∗ +Δ r ). (1.45) Following Takayama (1985), (1.45) holds if a positive answer is given to the following questions: 36 1. For any given c≥0, is there an b Δ such that b Δ= I− e X −1 c? Is such a b Δ unique? 2. Is the matrix I− e X nonsingular? If so, is it the case that I− e X ≥0? The following lemma is used in giving a positive answer to the above questions. One re- callsthatan× nmatrixBhasadominantdiagonaliftherearepositivenumbersd 1 ,d 2 ...,d n such that d j |b jj | > P i6=j d i |b ij | , for j =1,...,n. Lemma 1.1. The N× N matrix B := I− e X has a dominant diagonal. Proof: See Appendix. From theorems 4.C.3, 4.C.4 and 4.C.6 in Takayama (1985), I− e X havinga dominant diagonal implies that the answers to the existence and nonsigularity questions raised above are all positive and, not only that, I− e X −1 can be written as: I− e X −1 = ∞ X k=0 e X k =I + e X + e X 2 + e X 3 +.... (1.46) Combining (1.45) and (1.46), it then follows: e Δ=(Δ r ∗ +Δ r )+ e X(Δ r ∗ +Δ r )+ e X 2 (Δ r ∗ +Δ r )+ e X 3 (Δ r ∗ +Δ r )+.... (1.47) The terms in the series (1.47) can be interpreted in the following way: (Δ r ∗ +Δ r ) corresponds to the first wave of shocks banks receive from investments in small and large projects. Given the assumption that the networth of banks is negligible, every bank is bankrupt upon being hit by shocks of any dimension and, therefore, default by debtors on loans taken will ensue. Banks defaulting on loans will cause a second wave of shocks, e X(Δ r ∗ +Δ r ), that adds to the first one due to direct losses. The matrix e X provides the factors according to which a unit loss of debtors - column hea der banks - is spread among 37 creditors - row header banks. Multiplying e X by (Δ r ∗ +Δ r ), therefore, transforms the per unit losses in total losses. The second wave of shocks might lead to a third one in which, for example, a debtor bank1spreadslossestoacreditorbank2,whichinturnisadebtorofbank3,thislastonits ownbeingadebtorofbank1-inotherwords,ifthereisacolle ction ofbanksconnected ina circular chain. The interpretation of the additional terms in (1.47) is analogous and, as the simulations performed in the next section show, they tend to die fast, since in the majority ofthecases only ahandfulof banksconcentrate mostoftheloans madeintheeconomy. All the rest just make loans or invest in small projects, in this way not constituting channels for contagion that would give rise to terms of higher order in (1.47). The analogy with input-output analysis comes from the eleme nts of e X being viewed as the inputs necessary to produce the loss generated by banks, e Δ. For, from the analysis preceding (1.37) one knows that r i 1−α i r e Δ j /l j d represents the unitary loss produced by bank j and imposed on bank i - conditional on χ ij =1, i.e., bank i having lent to bank j. Therefore, e Δ j   r 1 1−α 1 r χ 1j l j d +...+ r N 1−α N r χ Nj l j d   (1.48) is the total loss produced by bank j. To make up such a loss one sums up its individual pieces, i.e., e Δ j " r 1 1−α 1 r χ 1j l j d # | {z } Bank j’s loss channeled to bank 1 +...+ e Δ j   r N 1−α N r χ Nj l j d   | {z } Bank j’s loss channeled to bank N , (1.49) so that, for any i, r i 1−α i r χ ij /l j d corresponds to the unit contribution of the loss imposed on bank i in the making of the loss produced by bank j. These input factors of production are elements of the j’s column in e X, as it would be in a standard input-output 38 matrix. 1.6 Measures of Financial Fragility The network structure automatically allows one to obtain the balance-sheet of banks, in which assets and liabilities can be characterized. The assets represent how banks are ex- posed to shocks, since they show the composition of portfolios in the different classes of investments, i.e., large projects, small projects, and loans. Liabilities might indicate how harmful a bank can be, since they detail to whom and in which amount banks owe money. The networth gives a measure of the whealth of a bank, since it provides what a bank has - assets - minus what it owes - debt. However important the networh of the financial system is, it does not capture entirely the exposureof a specific network of banks to shocks, since for contagion not only networth matters but also how banks are connected. In this way, this section develops two numerical measures of financial fragility based on the number of bank failures after banks’ assets are hit by shocks. In the first one, shocks are imposed on banks one at a time, allowing one to obtain which is the most critical bank to fragility in a specific network and, as it will be explained later, this exercise is analogous to the algorithm used by Google to rank webpages. In the second way of assessing fragility, all the banks are simultaneously hit by shocks, and in this exercise different distributions of parameters are used, in order to assess how the effects of intervention relate with the characteristics of the economy where intervention takes place. 1.6.1 A Simulation-Based Google Measure of Fragility As mentioned in the previous section, one approach to measure fragility is to consider, for a given network, the numberof bank failures following shocks on banks’ assets, one bank at 39 a time. This in practice constitutes a stress test of the network, in order to see how robust the overall structure is to problems its individual members might face. By considering the total cases of distress, direct and indirect, one takes into account contagion, as not only fails due to payoff shocks are considered but also those due banks’ default on their debt. Inthisway, consideranetworkwithN banks,andtakeanarbitrarybankifacingshocks in its projects. The set D i := n j6=i Δ j >W j o (1.50) contains all those banks that become distressed as a result of contagion, since the only bank facing direct shocks is bank i. The cardinality of this set, D i , thus, gives the total number of indirect failures following bank i’s shocks. For contagion to ensue, a necessary conditionisthatbankiisbankruptitselfand, therefore, if D i ispositive, thetotal number of bank failures is 1+ D i . The index f i :=        1+ D i if D i 6=∅ 0 otherwise (1.51) gives, therefore, a measure of the relative fragility of the network to bank i, for a particular realization of shocks that it faces. By doing the same for every bank j6=i in the network and combining all the results, i.e., taking f := X i∈N f i , (1.52) onehas ameasureof the overall fragility of thenetwork relative totheindividual failure of its members. To get more robust results, one perform multiple simulations drawing different shocks for every bank i∈N. To fix the idea, consider the following example of a network with 6 banks, generated 40 under ρ ∗ = 0.05ρ, γ ∗ = 0.8 and γ = 0.3, and its non-government counterpart, i.e., the network obtained in the same way but with γ ∗ = γ = 0. The other parameters used are given in Table 1.1: r ∗ r ω e Bank 1 3.23 1.19 0.05 0.04 Bank 2 3.00 1.01 0.17 0.12 Bank 3 2.22 1.19 0.15 0.11 Bank 4 2.55 1.08 0.09 0.07 Bank 5 2.97 1.00 0.15 0.12 Bank 6 2.71 1.21 0.14 0.10 Table 1.1: Parameters used for the generation of the networks in Figure 1.7. Network w/ Govt Intervention 1 2 3 4 5 6 Network w/ No Govt Intervention 1 2 3 4 5 6 Figure 1.7: Networks generated under the parameters of Table 1.1. Figure1.7gives thenetworks obtainedundersuchasetofparameters. Afterperforming 1,000 simulations for each bank, where both shocks hitting large and small projects, δ and δ ∗ , respectively, are drawn from independent U[0,1] uniform distributions, Table 1.2 gives the number of failures induced by shocks in each of the banks, split in indirect and direct cases of distress. Some other measures, like the leverage ratio, LR, and the number of (inward) links, are included. For the network obtained under intervention, calculating (1.52) one has f =5,159 and, 41 Government No Government Bank LR Links Ind Fails Dir Fails Networth Bank LR Links Ind Fails Dir Fails Networth 5.00 0.69 4.00 492.00 660.00 4.08 6.00 0.82 3.00 703.00 895.00 1.82 3.00 0.79 2.00 474.00 664.00 1.86 5.00 0.68 3.00 450.00 624.00 3.67 2.00 0.63 5.00 475.00 633.00 6.04 2.00 0.62 4.00 273.00 472.00 5.32 1.00 0.67 3.00 454.00 609.00 3.79 4.00 0.91 0.00 0.00 572.00 0.50 6.00 0.81 1.00 249.00 449.00 1.42 1.00 0.70 2.00 0.00 203.00 2.81 4.00 0.91 0.00 0.00 0.00 0.50 3.00 0.83 0.00 0.00 0.00 0.89 Table 1.2: Ranking of banks according to the number of failures induced. for its non-government counterpart, f =4,192, across a total of 100,000 simulations. If one is to consider these as measures of fragility, in this particular example the network under intervention would be deemed as more fragile. Both the total number of indirect and direct failures happens to be large in the govern- ment intervention network, so that even considering one of them instead of the total as a measure of fragility, it would still lead to the same result. However, this is not necessarily the case, given that intervention might increase the networth of banks and at the same time lead to a larger number of links, which means that in principle it would reduce direct failures and increase indirect ones. For any given network, a question that can be adressed using this methodology refers to which node is deemed to be the most important in terms of financial fragility, i.e., which is the bank that, once hit by shocks, will bring about the largest number of failures. An analogous questionistackled bysearch enginesintheinternet, wherethegoal isnottorank banks but, rather, to show the results of a particular query, in a way that links considered to be the most relevant - according to specific algorithms - ap pear first. PageRank, developedbyLarryPage, isthealgorithmusedbytheGoogleInternetsearch engine. Thealgorithmseekstorankahyperlinkedsetofdocumentsbymeansofassigningto each of them a measure, or weight, that summarizes their relative importance, as described in Chiang (2012). A network is a hyperlinked set, with edges, or links, going from nodes to nodes. ThePageRank algorithm assigns a weight to a nodeas afunction not of thenumber of nodesleadingtoitbut, rather, thenumberof nodesleadingtothenodesthatareleading 42 to it, and so on and so forth, in a recursive way. Such an idea finds its use in the way banks should be ranked in terms of their relative importance to financial fragility: for the same reason that a webpage with many links pointing to it might not be the most relevant in an internet query, a bank with the highest numberofconnections doesnotnecessarily needtobetheonewiththehighestpotential for causingfailures, thusnotthemostrelevantforfinancialfragility. Whatshouldmakeabank to be pivotal in a particular network is its own financial health - networth - taken together with that of the banks it is connected to. Contagion is easier to ensue in cases where weak banks operate together, since the failure of one will most probably bring about the failure of the other. This is what makes a variant of the PageRank algorithm to be useful in a ranking of banks where ones focus on their relative importance to financial fragility. Bysimulatingshocks,onebankatatime,andcalculatingthenumberoffailuresresulting from them, one analogously does for the network of banks what the PageRank algorithm does for the internet web. It is in this sense that f implicitly embeds on it the idea used by Google to rank webpages. It should not come as a surprise, therefore, that in a ranking of banks according to such a measure of fragility, the most important one will not necessarily be that with the largest number of connections, or inward arrows. Table 1.3 tabulates the percentual due to each bank for the number of indirect, direct and total failures. Government No Government Bank % Ind Fails % Dir Fails % Tot Fails Bank % Ind Fails % Dir Fails % Tot Fails 5.00 .2295 .2189 .2233 6 .4930 .3236 .3812 3.00 .2211 .2202 .2206 5 .3156 .2256 .2562 2.00 .2215 .2100 .2148 2 .1914 .1706 .1777 1.00 .2118 .2020 .2060 4 .0000 .2068 .1365 6.00 .1161 .1489 .1353 1 .0000 .0734 .0484 4.00 .0000 .0000 .0000 3 .0000 .0000 .0000 Table 1.3: Percentual over the number of indirect, direct and total failures. Going back to Table 1.2, the bank most highly ranked does not happen to be the one withthehighestnumberoflinks, whateverthepolicyunderconsideration. Intheparticular case of intervention, bank 5 lags behing bank 2 and, with no government, bank 6 is tied 43 with number 5 but has fewer links than bank 2. What makes bank 5 and bank 6 to lead the rankings for the two networks is that both are creditors of banks that have a relatively smallnetworthsothat, whenever5and6fail, anotherbankwillprobablybecomedistressed too. Bank 5 has a relatively high networth but not bank 6, which has a cushion to absorb shocks higher only when compared to that of the banks it is a debtor to - a typical example of weak banks operating together. Also, from Table 1.2 one notices that neither bank 5 or 6 has either the highest leverage ratio, LR i , given by LR i :=1− W i a i = Total debt z }| { l i ω +l i d a i |{z} Total assets , (1.53) or the lowest networth across the banks in the network they belong to. For instance, bank 5 is only the 4th most leveraged bank, and has the 2nd highest networth, whereas bank 6 occupies the 3rd position in terms of leverage and the 4th in terms of networth. This shows how important the network structure turns to be if one aims at spotting those banks that can cause a great deal of a problem once they become distressed. Focusing only on more traditional financial indicators, like leverage and networth, might be misleading, and for sure it is in the example at hand. 1.6.2 Financial Fragility and Distribution of Parameters However insightful it might be for the study of financial fragility and in determining the relative importance of banks, it is also plausible to have shocks hitting the entire network, and not only banks one at a time. In a situation like that, instead of a single bank being the source of distress, every bank becomes part of the problem as soon as they all are hit by shocks. Not all will fail and spread contagion, but some might do only because of third parties defaulting on their debt. In this regard, consider the following sets: 44 D := n i∈N Δ i >W i o (1.54) and D ′ := n i∈N Δ i r ∗ +Δ i r >W i o . (1.55) Set D contains all those banks that fail as a result of either direct shocks or contagion, whereasD ′ hasinitthosethatwouldbedeemedbankruptevenintheabsenceofcontagion. It follows that D ′ ⊆D, allowing a third set to be defined, D ′′ :=D\D ′ , (1.56) which in turn contains only banks that fail due to contagion. Analogously to (1.52), f :=|D| = D ′ + D ′′ (1.57) is a measure of financial fragility capturing how banks are exposed to direct shocks and indirect failures, due to them belonging to a particular network. Contrasting f obtained in networks with and without intervention, one gathers how important government’s policy is for financial fragility. Also, it is necessary to split the total number of bank failures in cases of direct and indirect distress, since intervention might affect these numbers in opposite directions. For instance, intervention might increase network’s networth, leading to a smaller number of direct bank failures but, since it might also increase the number of links, it can make banks more prone to become distressed due to contagion. Theframework developed hasmany parameters, andonequestion thatarises is how the effects of intervention are related to the characteristics of the economy where intervention takes place. To make a differentiation in the set of parameters that would give rise to 45 distincttypesofeconomies,oneusesdistributionsthatdifferintermsoffirstorderstochastic dominance. Thedifferenceamongtheseeconomiescouldbethoughtofasdifferencesintheir respective levels of financial development, as it is explained in the sequence. 1.6.3 Simulations and Financial Development In the framework developed, government intervention is a way of allowing banks to recoup a fraction of their costs, due to the premature selling of projects, which in turn is caused by the maturity mismatch between assets and liabilities in their balance-sheets. Instead of running simulations and relating changes in the outcomes from government’s policy to changes in the parameters, one could additionally impose a finer partition on the support of the parameters’ distributions and associate to these different partitions different types of economies. These economies could be thought of as having distinct levels of financial development, which would provide a way of looking at how financial development relates to the effects of government intervention. Recall that the parameters of the model are: • r ∗ : return on large projects; • r: retutn on small projects; • ω: fraction of early depositors; • e: equity (capitalization); • ρ and ρ ∗ : fire-sale parameters of small and large projects; and • γ and γ ∗ : government intervention in small and large projects. The economies that are to represent different cases of financial development are labeled high (h), middle (m) and low (l) level of development economies. They are distinguished according to the following assumptions: 46 1. The distribution of payoffs across the economies is ranked as: U r ∗ l fosd U r ∗ m fosd U r ∗ h (1.58) for large and U r l fosd U r m fosd U r h (1.59) for small projects, i.e., the distribution of projects’ payoff of a low development econ- omy first-order stochastically dominates that for a middle o ne, which in turn dom- inates that of a high development economy. Implicitly it is assumed that in less developed economies projects paying high payoffs are yet to be explored, and the opposite for less advanced economies; 2. For the distribution of early depositors: U ω l fosd U ω m fosd U ω h , (1.60) i.e., depositors in low development economies are more predisposedto withdraw early as they are supposedly less equipped to absorb liquidity shocks; 3. Banksare, onaverage, lesscapitalizedthelowerthedevelopmentleveloftheeconomy: e h >e m >e l . (1.61) Regarding the other parameters, one assumption previously made was that the prof- itability of small projects, expression (1.22), is nonnegative, otherwise banks would not even be willing to take deposits from retailers. In this way, for any specification of r ∗ , r, w, 47 eandγ, thefire-saleparameter ρisendogenously determinedsothatnobankisataloss by going small. Having ρ endogenously determined, the fire-sale parameter for large projects, ρ ∗ , is set so that it represents a fraction of the one for small projects. In the simulations performed, that captures circumstances when having to sell a large project before maturity gets more and more costly relative to selling a small one. By setting the parameters as above one can make an inter comparison of economies and see how the effects of government’s policy on them differ. One is also interested in seeing how different policies affect the same economy, i.e., an intra comparison. For that, four different levels of government intervention for large projects are considered, namely high, γ ∗ h , medium, γ ∗ m , and low, γ ∗ l , the last one representing the case where the level of intervention in large and small projects is the same. These are set as: γ ∗ h >γ ∗ m >γ ∗ l =γ. (1.62) Theparametercorrespondingtogovernmentintervention forsmallprojects,γ,isinturn simulated at three different levels, high, γ h , medium, γ m , and low, γ l , with: γ h >γ m >γ l . (1.63) Table 1.4 gives the specifications used to generate the set of parameters for the simula- tions: • ρ ∗ /ρ: gives the relation between ρ ∗ and ρ, the fire-sale parameters for large and small projects, respectively, showing how costly it is a premature selling of a fraction of the large project relative to the cost for a small one. In particular, for ρ ∗ /ρ = .05, a fraction of the small project at t = 1 is worth only ρ whereas for a large project it is only 5% of that, i.e., ρ ∗ =.05ρ (recall that ρ is endogenously determined); 48 • γ: percentual of government subsidy for small projects; • γ ∗ h , γ ∗ m and γ ∗ l : percentual of government subsidyin varying degrees of intensity, as in (1.62). For a high level of intervention, γ ∗ h = γ ∗ , for a medium level of intervention, γ ∗ m =γ+2 γ ∗ −γ /3, and for a low level of intervention, γ ∗ l =γ, where γ ∗ =0.80; The simulations consider a network of N = 10 banks, and the shocks on the payof of small and large projects, δ and δ ∗ , respectively, are independently drawn from uniform distributions U[0,1]. Simulation ρ ∗ /ρ γ γ ∗ h γ ∗ m γ ∗ l I .05 .30 .80 .63 .30 II .10 .30 .80 .63 .30 III .15 .30 .80 .63 .30 IV .20 .30 .80 .63 .30 V .05 .50 .80 .70 .50 VI .10 .50 .80 .70 .50 VII .15 .50 .80 .70 .50 VIII .20 .50 .80 .70 .50 IX .05 .70 .80 .76 .70 X .10 .70 .80 .76 .70 XI .15 .70 .80 .76 .70 XII .20 .70 .80 .76 .70 Table 1.4: Parameters used in the simulations performed. For each possible level of intervention, the parameters are drawn 10,000 times, for each type of economy. Since there are 3 of those levels, namely γ ∗ h , γ ∗ m and γ ∗ l , and three types of economy, every simulation involves 90,000 draws from uniform distributions as specified below. Letk =1,2,3denotethehigh,middleandlowdevelopmenteconomies, respectively. With r ∗ = 6, r = 2 and ω = 0.50, the parameters for the simulations are generated in the following way: • Payoff of large projects are drawn from U r ∗ k :=U[r,r ∗ k ], (1.64) 49 with r ∗ k =r+k(r ∗ −r)/3; • Payoff of small projects from U r k :=U[1,r k ], (1.65) with r k =1+k(r−1)/3; • Fraction of early depositors from U ω k :=U[0,ω k ], (1.66) with ω k =kω/3; • Capitalization of banks is taken to be e k = ω k k+1 ; (1.67) where ω k is drawn as previously for economy k. With the parameters in place, one obtain network structures for the high, middle and low development economies. Coupled with these are the networks obtained under the same parameters but whereintervention is nonexistent, after all the objective is to determine the effects of the presence of the government. For that, as mentioned earlier, the metric to be used is the total number of defaults, f, as defined in (1.57). That is obtained by imposing the shocks on small and large projects for each bank, in every network, and calculating how that spreads to other banks. The number of failures is obtained and averaged out, after the simulations are run for each specific type of economy and for each specific level of intervention. The results obtained are analysed in the sequel. 50 1.6.4 Qualitative Results from Simulations Some qualitative statements can be made upon the results obtained from the simulations. The first one is: Government intervention leads to more financial fragility. The level of financial fragility in economies under no intervention is persistently higher compared to those in networks obtained without the participation of the government. For low development economies, the financial fragility indicator is at least 40% higher when the government is present. The difference turns to be small only when intervention does not change much the recovery rate of projects, which is the case when one has in the same scenario a high development economy and a relatively small fire-sale cost. The number of direct bank failures is not as much affected by government intervention as is the number of indirect cases of distress. Intervention increases both, for all types of economies, but the number of indirect bank failures associated with structures under participation of the government is significantly higher. The following is a result related to the higher number of indirect failures under govern- ment intervention, which can be viewed as a numerical validation of Proposition 1.1: Government intervention leads to a persistently higher number of links. Indirect bank failures are dueto contagion, when otherwise healthy institutions become distressed as a result of third parties not fulfilling their debt obligations. Therefore, a necessary condition for indirect failures is the existence of links across institutions, since those represent loans taken as a way of financing large projects. Proposition 1.1, by saying that intervention does not brake links across banks - althou gh it might change the identity of the borrower and the lender - thus helps in the understandi ng of why network structures 51 generated under the government’s presence have a higher number of indirect failures - basically, because they have more links. The next result is a consequence of Corollary 1.2, and it has also an impact on the number of bank failures: Government intervention leads toapersistently higher levelof total networth. This, in theory, should partially offset the increased exposure of banks to third part failures discussed above, since networth is the cushion that banks have to absorb shocks. On average, the simulations show that total networth under government intervention is persistently higher than without, for all types of economies, and in particular for the less developed types. Thus, one in principle would expect networks under intervention to be associated with a smaller number of failures. What happens is that, even though increasing the total networth of banks, intervention makes them to diversify less their risks, since banks that otherwise would be investing in small projects - in their respective regions and with their r espective idiosyncratic shocks - now concentrate their risks in the very same institution to which loans are made. When an institutionthatbecomesadebtortomanyothersishitbyalargeshock, itshighernetworth - coming from investments in large projects - is not enough to prevent the shock it receives from rippling across the network. Forthecreditors,thus,itbecomesasifbeinghitbythoseverysamelargeshocks. Recall that banks’ projects are hit by shocks coming from independent uniform distributions, so that the probability of a single bank being hit by a large shock is higher than that of two or more being hit at the same time by a corresponding large shock. This is what makes a higher number of banks to succumb to large shocks when intervention is in place. All the more surprising is the fact that, for the very same type of economies where it is verified the highest increase in networth upon intervention - low development economies 52 - it is also observed the highest increase in the level of finan cial fragility. The way these economies are designed - having a higher fraction of early de positors and beingless capital- ized-makestheirstructuretobeinherentlymorefragilean dconcomitantlymoredependent on government’s policy, and that is why the results become much more significant for them in a government versus non government comparison. One traditional financial indicator that is often looked upon as a way of measuring the degree of exposure to shocks is the leverage ratio. The higher the leverage ratio, the higher is the fraction that debt represents of total assets and, therefore, the more trouble would have a bank to fulfill its promises as soon as it finds itself in financial hardship. The next result relates the impact of government intervention on the leverage ratio: Government intervention leads to no significant changes in the average lever- age ratio of a network. The simulations show that intervention does not have significant effects in the leverage ratio, regardless of the economy’s type. The reason is that, in the framework proposed, government leads toan almost proportional increase in bothassets and networth and, thus, the leverage ratio barely changes, even though assets and networth do. This result, combined with the fact that intervention brings more fragility, shows how misleading it can be if one looks only to leverage as a measure of the healthiness of the financialsystem. Inotherwords,itshowshowimportantknowledgeofthenetworkstructure and the bilateral exposure of banks is in determining financial fragility. Another result from the simulations concerns the effects of different levels at which the government implements its intervention policy. Using the too-big-to-fail terminology to designate those instances where the support offered by the government for large projects is higher than the equivalent one for small projects, one has that: A too-big-to-fail intervention policy does not have much im pact on either 53 financial fragility or total networth. Despite financial fragility being quite the same under different intervention levels, it is still persistently higher for low development economies. The same does not hold for total networth,asthedifferenceacrosseconomiesgetsflatasthefire-salecosttoselllargeprojects becomes less severe. Thatmeans that, unlesstherecovery rateof large projects is relatively low, if the aim of the government is to increase banks’ total networth, then offering more subsidies to large projects is a waste of money and, in case one deals with a low developed economy, it will come at the expense of more financial fragility. Oneparameterthatwasvariedacrossthesimulationsisthefire-salecostoflargeprojects relative to that of small ones. Regarding that, the results show that: Government intervention has a more significant impact on financial fragility and banks’ total networth when the fire-sale cost of large pro jects is high. This result is due to the fact that the fire-sale cost is one of t he main drivers - together with government’s policy - of the recovery rate of projects. When the fire-sale cost of large projects is relatively small, government intervention might notbeenough to tip thebalance in favor of large projects. This in turn makes the number of links in government versus non-government networks to be similar, and thus the impact o f intervention on financial fragility and total networth vanishes, since these are mainly dependent on the number of links. As already mentioned, one should keep in mind that it is not the case that intervention becomesirrelevantwhenthefire-salecostoflargeprojects isrelativelysmallbut,rather,that it becomes less significant. For instance, in the simulations for low development economies, one still gets a level of financial fragility more than 40% higher with intervention than without, even with a low fire-sale cost. 54 1.7 Concluding Remarks This paper studies a model of endogenous formation of the network of banks, where gov- ernment intervention plays a key role in determining the structure of the banking system. Government’s policy, i.e., intervention, leads to the formation of different networks, and since each of them presents a distinct degree of financial fragility, one can analyse the relationship between intervention and financial fragility. To analyse financial fragility, the measure defined is the number of bank failures after shocksareimposedonbanks’assets. Thismeasureiscomputedintwodifferentways: inthe first banks are hit individually, allowing one to obtain which of them exposes more the net- worktofragility, inanexerciseanalogoustothealgorithmusedbyGoogletorankwebpages. Second,banksarehitsimultaneously,andsimulationsarerunusingparameterscomingfrom distributions having varying degrees of dispersion, in terms of first order stochastic dom- inance. These different distributions lead to the characterization of economies that could be thought in terms of financial development, and by doing that one can analyse how the effects of intervention interact with the characteristics of the economy where intervention takes place. In terms of theoretical results, it is shown that intervention makes the network of banks to bemoreconnected, andthe moreintensetheparticipation of thegovernment thegreater the connectivity of the network will be. Also, despite subsidising banks, intervention does not necessarily lead to an increase in the wealth of the banking system. In terms of simulation results, for the case when shocks are imposed on banks one at a time it is shown that the most critical bank as far as fragility is concerned is not necessarily the biggest, the more leveraged or the more connected. What makes a bank to be pivotal is its own criticality and its connectivity with other banks that are also deemed fragile. For the case when all the banks are hit simultaneously, it is shown that intervention brings 55 more financial fragility, even though it increases the wealth of the banking system, and this effect is stronger in those economies where the parameters are more dispersed: the ones labeled low development economies. Also, vis a vis a network without the government, intervention does not affect the average leverage ratio of the financial system, even though causingasignificant changein thestructureofthenetwork and, consequently, inthedegree of fragility. Therefore, the framework studied highlights the importance of one’s knowledge about the bilateral exposureof banks, on top of more traditional measures like the leverage ratio. The two main testable implications of the model are that (i) intervention increases the connectivity of the network of banks and (ii) intervention brings more financial fragility, despite increasing the wealth of banks. One could test these two hypotheses by using data measuring the bilateral exposure of banks in countries with presumably different levels of government participation in the market for distressed assets, and see how the banking system is affected during crises episodes. Government intervention is modeled in a reduced form, without consideration of how its subsidyfor banks is financed. This is an important issue that needs to beaddressed in a future version of the model, in order to capture general equilibrium effects of government’s policy. Also, intervention as defined in the model creates moral hazard in the banking system, in the sense of inducing banks to invest in less liquid assets. This is an interesting aspect of the model that deserves to be further investigated in future research. 56 Chapter 2 Government Safety Net, Stock Market Participation, and Asset Prices Thischapter studiestheeffect on equilibriumprices adventing from thepresenceof a safety net during financial crises. It is shown that, by inflating prices with the insurance provided through its intervention policy, a government might be sowing the seeds of a crisis that its intention is to prevent in the first place. The model developed is one with risk-neutral agentsfacingastaticdecisionproblem,underdifferent(i)frameworks-withandwithoutthe possibilityofintervention -and(ii)informationalscena rios-imperfect,perfectandcommon priorinformation. Intervention occurswheneverthecombinedreturnofthoseinthemarket goes below a certain threshold. By having different frameworks and scenarios, the impact of the safety net on a diverse class of assets can be studied. Equilibrium prices are shown to be always higher in the framework with the possibility of intervention, regardless of the informational scenario. There is a limit on price inflation, however, since an equilibrium fails to exist in those instances where high prices indicate an intervention to be imminent. 57 The effect on prices of an increase in the degree of uncertainty turns out to be dependent on the supply level of the asset. 2.1 Introduction During crises episodes, either financial or economic related, the course of action of the government is always subject to a great deal of controversy. No consensus is ever achieved between those who, on the one side, champion the idea of government intervention and those who, on the other side, believe in the self-correcting capacity of markets. From an ex post point of view, government intervention might be required, if the state of affairs is not to be aggravated; from an ex ante perspective, if government intervention is taken for granted whenever bad outcomes happen, risks might be incurred in excess. The way the government behaves in crises episodes has an impact on the payoff of virtually any asset. Thereasonis that, atleast to somedegree, thereis always acorrelation between an asset’s payoff and the state of fundamentals of the economy - those factors that indicate how well the economy is performing, and the very same factors the government aims at when intervening. Affecting the payoff, the government turns out to be important for the investment decision of agents, which in turn affects the demand for the asset and, consequently, its price. The asset’s price is part of the state of fundamentals, causing the action of the government to feed back into itself. Figure 2.1 illustrates this process. An example of the aforementioned process is the case of mortgage-backed securities (MBS)in thefinancial crisisepisodeof 2007-2009. MBSarese curitites whosepayoff derives from a pool of mortgages, assembled together an issued as a single asset, in what is known as securitization. Securitization basically creates a secondary market for loans, which helps in channeling the necessary funds for financial institutions to create new mortgages. Of prominent role in this secondary market is Fannie Mae, a company created by the 58 Asset Payoff Asset Price Government Demand 4 2 3 u u u u 1 Figure 2.1: Impact of government intervention and feedback. US government in 1938, with the goal of fostering the level of home ownership in the country. Initially established as a government-sponsored enterprise (GSE), itwasconverted into a publicly traded company in 1968. This change of ownership altered its government guarantee status, from being explicit pre-1968 to being implicit pos-1968. Quoting the Wikipedia entry for Fannie Mae 1 , Originally, Fannie had an “explicit guarantee” from thegov ernment; if itgot introuble,thegovernmentpromisedtobailitout. Thischangedin1968. Ginnie Mae was split off from Fannie. Ginnie retained the explicit guarantee. Fannie, however, becameaprivatecorporation,withonlyan“implie dguarantee”. There was no written documentation, no contract, and no official promise that the government would bail it out. The industry, government officials, and investors simply assumed it to be so.(...) This implied government guarantee would constitute arc number 1 in Figure 2.1. Since the payoff of MBS are attached to the payment of the loans themselves, it is an instrument with a great deal of credit risk. Therefore, the support from the US government conferred to the securities traded by Fannie Mae a lot of appeal, increasing the demand for them - arc number 2. From Wikipedia, 1 http://en.wikipedia.org/wiki/Fannie_Mae. 59 However, the implied guarantee, as well as various special treatments given to Fannie by the government, greatly enhanced its success. As the story goes, with an active secondary market for loans - and a fierce competition fornewcustomers, inabusinessdeemedprofitableatthetime-aplethoraof creditbecame available to those willing to assume a mortgage. With the availability of credit came a decrease in the lending standards, setting the stage for a disaster: the increased demand for houses inflated a bubble - the boom, arc number 3 - and, when the high level of prices could not be sustained anymore - the bust - those disqualified borrowers had no ways of fulfilling their obligations. By the time, a great fraction of the mortgage market was owned by Fannie Mae, whose collapse would pose a serious threat to all those invested in assets like MBS - making government intervention inevitable, arc n umber 4. (...)Fannie and Freddie underpinned the whole U.S. mortgage market. As recentlyas2008, FannieMaeandtheFederal HomeLoanMortgageCorporation (Freddie Mac) had owned or guaranteed about half of the U.S.’s $12 trillion mortgage market. Iftheywereto collapse, mortgages would behardertoobtain and much more expensive. Fannie and Freddie bonds were owned by everyone from the Chinese Government, to Money Market funds, to the retirement funds of hundreds of millions of people. If they went bankrupt, all those investments would go to 0, and there would be mass upheaval on a global scale.(...) As far as government intervention is concerned, not only the payoff of assets in a partic- ular segment - e.g., mortgage securities held by Fannie Mae - but also the payoff of assets associated with a broader measure of the performance of the economy - e.g., stock market indices - matter. For, the latter has the potential of draggi ng the economy down - through the expectations channel - that is not necessarily present in the former. If th e Dow Jones index succumbs to a large fall, for example, the potential losses that might result from 60 the economy dipping into a recession - a situation that could arise as investors lose confi- dence and withdraw from the market -pose a threat serious eno ugh to demand government intervention, regardless of how important the realized losses are. In this way, government intervention can be associated with different classes of assets - all that matters is them being important from a systemic poi nt of view, posing a risk that cannot be diversified away by the individual action of agents. The overall impact of the presence of the government - on the behavior of investors and, therefore, prices - is determined by the interaction of the characteristics of assets in different classes with that possibility of intervention. For, just to pick two possible dimensions, consider (i) payoff uncertainty and (ii) overall level of investment in the asset. Regarding (i), investors might have a varying degree of information across assets in different classes - less information about the likes of mortg age securities, more information about the likes of the Dow Jones index. The degree of uncertainty associated to the payoff of an asset being higher, so more valuable one would expect the possibility of government intervention to be, as investors would have moresafeguards against those not contemplated cases where the asset performs extremely poorly. With respect to (ii), the more pervasive is the presence of the asset in the economy, the more one would expect it to contribute to systemic risk, making the possibility of an intervention more valuable too. After all, were the asset to represent only a small fraction of the market, an intervention would not make much of a difference, unless for the handful of agents invested in it. Implications like those outlined above, resulting from the possibility of government in- tervention and being reflected in the behavior of investors and prices, constitute the focus of the paper. The main goal is to study how safeguards from the government interact with different characteristics of assets - in particular, uncert ainty - as far as the determination of equilibrium prices is concerned. With that, those effects that are a byproduct of the 61 tension between creating more incentives for risk taking and preventing the economy from getting into a worse situation - tension which is inherent in the intervention decision of the government - can be analysed in more solid grounds. In the setup to be presented, the measure that the government looks at when deciding to intervene is the total welfare of investors, to be called social welfare - defined as the sum across agents of thereturnon their portfolios. Whenever thatmeasuregoes below a critical level, the government intervenes and social welfare is restored to that level 2 . Taking the possibility of government intervention into account, investors are asked to form their portfolios. First, agents are risk neutral, with utility adventing only from the proceeds of the investment strategy. Second, investment can be made in two assets, a riskless and a risky one. The riskless asset pays no interest and only allows agents to transfer wealth from the initial to the final period of the economy, whereas the risky asset has a payoff which is perfectly correlated with the state of fundamentals. The state of fundamentals is modeled as a uniform random variable, the realization of which the government is assumed to know. To capture the variation in the degree of uncer- taintyattached todifferentassets, investorsareplacedinoneofthreepossibleinformational scenarios, namely (i) imperfect information scenario, where each investor receives a private signal of the realized state of fundamentals, (ii) perfect information scenario, where every agentknowswhatthestateoffundamentalshappenstobe,and(iii) common priorscenario, where all the agents know is the probability distribution of the state of fundamentals. To explore the consequences of government intervention, two frameworks are studied: one where investors entertain the possibility of intervention - labeled government interven- tion-andanotherwhereinvestors ruleoutthatpossibilityfrom theoutset- nogovernment. To see how the possibility of intervention affects different types of assets - as far as uncer- 2 The paper abstracts from how government intervention is implemented, e.g., a decrease in the interest rate or a bailout of a specific institution. All that matters is that intervention results in the social welfare being set at the critical level. 62 tainty isconcerned-foreach oftheframeworkstherearethr eepossible scenarios, regarding the information of agents and labeled as such, following the above - imperfect information, perfect information and common prior 3 . As indicated by the diagram of Figure 2.2, for each possible combination of framework and scenario, the equilibrium price is derived. Then, for a given framework, the prices that prevail under each of the scenarios are compared - a comp arison intra framework, inter scenario. In a similar fashion, for a given scenario, the prices that result under each of the frameworks are related - a comparison inter framework, intra scenario. As far as the ordering of equilibrium prices is concerned, the former provides a way of analysing the effects ofuncertainty-foragiven level ofgovernmentinter vention -whereasthelatterfocus on the effects of the possibility of government intervention - for a given level of uncertainty. Government NoGovernment ImperfectInformation PerfectInformation CommonPrior ImperfectInformation PerfectInformation CommonPrior IntraFramework, InterScenario InterFramework, IntraScenario Figure 2.2: Effects from government intervention. 3 From now on, framework refers always to the presence or not of the government, whereas scenario is related to the degree of uncertainty faced by investors. 63 In the setup considered, a prerequisite for an equilibrium to exist is the market clearing condition being satisfied, i.e., the supply being equated to the demand. At the equilibrium price, the mass of investors willing to buy matches the quantity of the asset available - any deviation from that is a sign that the price is either too high or too low. Since investors know that (i) the government steps in only if social welfare falls below a critical level and (ii) the social welfare is roughly given by the total supply of the asset multiplied by the difference between its payoff (revenue) and its price (cost), that very same price at which investors are offered the asset indicates how likely the government is to intervene. By forming beliefs of the likelihood of an intervention, investors can calculate the expected returnonanyportfolio, whichiscrucialsince, withriskneutralpreferences, allthatmatters is the expected return - as far as the decision of choosing a po rtfolio is concerned. Regarding the portfolio decision of agents, they are allowed to invest as much as they want in the riskless but not in the risky asset, conditional on their budget constraint being satisfied. The reason is that, rather than exploring the impact on prices of the trading activityofaparticularinvestor-particularitythatcoul darisefromanagentbeingwealthier or better informed than the others - by modeling the investme nt on the risky asset as a “buy” or “not buy” decision, one can focus on the effects of age nts’ overall participation in the market, which can be as important a driver of prices as wealth or information 4 . As it turns out - manifested in the expression of the equilibr ium prices - the downside risk is one of the crucial factors determining the investment decision of agents. For, since utility derives only from net investment proceeds, agents need to contemplate how much of a relative loss a worst case-type of situation would entail, were an investment in the ri sky asset to be made. By relative loss it is meant the difference between (i) the total wealth that could be secured for consumption through a fully investment in the riskless asset and 4 This point is debatable but it is the view the paper takes on the issue, nonetheless. One could well argue that, for some types of assets, e.g., illiquid ones, trade is motivated primarily by information, so that, price-wise, the behavior of a well informed investor would b e more relevant vis-a-vis the overall level of investment. 64 (ii) the level of consumption that would result in the aforementioned worst case situation - by definition, that where the government is required to intervene. This is precisely what a measure of the downside risk captures. Since government intervention results in the state of fundamentals being set at the critical level, which is higher than what would prevail otherwise, and, being the state of fundamentalsperfectlycorrelatedwiththepayoffoftheriskyasset, interventioncanbeseen as an insurance - or, as a safety net - provided by the government. Intrinsically associated with this insurance is, therefore, the downside risk: the higher the former, the lower the later. Different assetswithdifferentlevels ofdownsideriskcan beinterpreted asassets with different levels of government guarantee. Not only those factors directly involved in the definition of the downside risk per unit of the asset - investors’s individual wealth, government’s insuranceor safety net and supply level - but also others like the state of fundamentals, range of uncertainty and transaction cost matter for the determination of equilibrium prices. They appear in different ways and, as such, have different implications for the comparative statics of the equlibrium price, in accordance with the framework / scenario under question. In terms of results, following Figure 2.2, a comparison of equilibrium prices inter frame- work, intra scenarioreveals that(i) thepossibility of government intervention hasundoubt- edly a positive effect on prices: no matter what the level of uncertainty, prices under the possibility of intervention are at least as high as those that prevail when government is absent, whereas a comparison of equilibrium prices intra framework, inter scenario shows that (ii) the possibility of government intervention does not alter the effect of uncertainty on prices: overall, with or without the possibility of intervention, prices in different infor- mational scenarios obey a certain ordering, in accordance with the realized state of funda- mentals. A result that holds under any framework / scenario specification is that (iii) no equilibrium can be sustained when investors are certain about government intervention. 65 Looking specifically to the framework with the presence of the government, the equi- librium derived in the imperfect information scenario shows that (iv) the equilibrium price increases with uncertainty for assets in low supply, but decreases otherwise, whereas that very same equilibrium price is (v) increasing in both the state of fundamentals and the government’s safety net, while dereasing in the supply level, the individual wealth of in- vestorsandthetransactioncost. Intheperfectinformationscenario,thecomparativestatics are straightforward: (vi) the equilibrium price depends only on the state of fundamentals and the transaction cost, increasing in the former and decreasing in the later. Finally, in the common prior scenario (vii) the equilibrium price depends on the individual wealth of agents: for suficiently low / high levels, the wealth constraint is binding and the price that prevails equals that individual wealth level, whereas for intermediate levels the equilibrium priceisincreasinginboththesupplylevel andthegovernment’s safetynet, whiledecreasing in the individual wealth of agents and the transaction cost. In the framework with no government, for the scenario with imperfect information, the effects of changes in the degree of uncertainty are also dependent on the supply level of the asset, as in the framework with the possibility of intervention: it is shown that (viii) the equilibriumpriceincreaseswithuncertaintyforassetsinlowsupply,butdecreasesotherwise, whereas that very same equilibrium price is (ix) increasing in the state of fundamentals, while dereasing in the supply level and the transaction cost. In the perfect information scenario, theresultisthesameoneobtained inthepreviousframework: (x)theequilibrium price depends only on the state of fundamentals and the transaction cost, increasing in the former and decreasing in the later. Finally, in the common prior case, (xi) the only variable that enters the expression of the equilibrium price is the transaction cost of the asset: investorsarewillingtopaytheunconditionalexpectedpayoff oftheriskyassetminus the transaction cost that they are charged when the liquidation of the portfolio takes place. 66 2.1.1 Related Literature If not directly studying the impact on prices, other papers also approach the implications adventing from government intervention in crises episodes 5 . Farhi and Tirole (2010) show that a policy whereby the government responds to crises decreasing the interest rate leads banks to choose a higher leverage ratio, with government intervention and maturity trans- formation being strategic complements. This implies that, with government intervention, the degree of maturity mismatch in the economy increases, leaving it more exposed to liquidity shocks. In Diamond and Rajan (2011), however, the very same interest rate policy - an undi- rected intervention - turns out to be better than the alternatives, in particular a bailout of a specific institution - a direct intervention. The interest rate policy preserves the pri- vate commitment of banks with their depositors, a point stressed by Diamond and Rajan (2001). Relative to the natural equilibrium rate, since banks know that the interest rate will be lower at times of stress - when needs are high - in order to discourage the very same banks to make commitments that increase the need for intervention, the interest rate needs to be higher in normal times - when needs are low - which r equires a credibility in a new direction, a term coined by the authors. Without this, a central bank might actually increase the likelihood of booms and busts. Still related to the portfolio choice of financial institutions, Acharya, Shin, and Yorul- mazer(2011)askthequestionofhowbanks’exantechoiceofliquidityisaffectedbydifferent governmentpolicies: (i)bailouts,(ii)unconditionalliquiditysupporttosurvivingbanksand (iii) liquidity support to surviving banks conditional on the level of liquid assets in their portfolios. Thefirsttwo policies are showntoreducebanks’ incentives to holdliquid assets, whereas those incentives increase with the third. 5 Thewordgovernmentinthepresentworkisusedinabroadsense,beingunderstoodeitherasacountry’s Treasury or its Central Bank. 67 Acharya and Yorulmazer (2007) analyse the implications of government intervention policies that are designed to be implemented only in systemic crises - those that affect a significant portion of the banking industry. If that is the case, banks are induced to herd, i.e., correlate their portfolios, since only when they fail together they will be bailed out - otherwise they are just liquidated. This results in a too-many-too-fail -type of guarantee by the government, exacerbating the number of systemic banking crises. Interestingly, differentlyfroma too-big-too-fail policy, alsosmallinstitutionsaresusceptibletothiseffect. Among others, Ennis and Keister (2009) study the ex ante effects of government inter- vention in the context of bank runs. The government policy that would rule out the run equilibrium in the Diamond and Dybvig (1983) setup, namely a deposit freeze, is shown to beex post inefficient, i.e., once arunon abankisunderway, prohibitingwithdrawals is not the best course of action for the government - the classic time-inconsistency problem , Kyd- land and Prescott (1977). Anticipating that, depositors have incentives to participate in a run once one takes place, implying that ex post efficient policies might have a destabilizing effect. Thepresenceofalenderoflastresort,e.g. theIMF,isanotherinstancewheretheprivate motives of agents might be distorted, eventually leading to moral hazard. In situations where a country is illiquid, Morris and Shin (2006) identify the circumstances under which an IMF bailout would convince investors to roll-over their m aturing claims and also the country beinghelped to engage in thecostly adjustmenteffort. Similar results are obtained byCorsetti,Guimarães,andRoubini(2006), wherethetrade-offbetweenbailoutsandmoral hazard is also central. Onepaperthat addresses theissue of investors’ participation in the market is Allen and Gale(1994). Inordertoexplainasset-pricevolatility, Al lenandGaledevelopamodelbased on incomplete market participation and heterogenous liquididy preferences of investors. According to their theory, a fixed setup cost might prevent some investors from trading 68 in a market so that, when a liquidity shocks hits the economy, it cannot be absorbed without causing a large change in prices. In their model, limited market participation is caused by the heterogeneity in investors’ liquidity preferences whereas in the present paper the driver is the heterogeneity in the information received: when investors have imperfect information, only those with a favorable outlook on the asset payoff will choose to participate in the market. Similarly to their model, however, is the result that different equilibria with different degrees of investors’ participation can be accomodated, resulting in diverse equilibrium prices. The structure of the paper is as follows: section 2.2 introduces the model, with the correspondingmeasure of social welfare used by the government to decide whether to inter- vene or not in the economy; section 2.3 focus on the framework where investors entertain the possibility of government intervention, with the following subsections dealing with the different informational scenarios: imperfect information, perfect information and common prior; analogously, section 2.4 refers to the framework where investors acknowledge the absence of the government; section 2.5 compares the equilibrium prices across frameworks (with or without the government) and across scenarios (imperfect, perfect and common prior information); section 2.6 concludes. The derivation of some of the results is contained in the appendix. 2.2 Model The model is composed of a continuum of agents, represented by the unit mass interval, I =[0,1], facing a static decision problem. There are four dates, t=0,1,2,3: • At t=0, the state of fundamentals is drawn, with e θ∼U[0,1]; • At t=1, agents decidewhether tobuyor not, X i =1or X i =0, respectively, asingle unit of an asset that has a payoff equal to the realized state of fundamentals, with 69 the degree of information about the asset’s payoff depending on the informational scenario: – Imperfectinformationscenario: investorsreceiveasignalabouttherealizedstate of fundamentals; – Perfect information scenario: investors are informed of the asset’s payoff; – Common prior scenario: investors only know the probability distribution of the state of fundamentals, assumed to be common knowledge. • Afteragentsmaketheirinvestmentdecisions,att=2thestageissetforthepossibility of government intervention, which depends on the framework: – Government framework: the government anticipates the social welfare level (to be defined) that results from investors’ strategies and decides whether or not to intervene; – No government framework: no intervention ensues, regardless of the social wel- fare level, with the third period being just a convention. • At t = 3, agents liquidate their portfolios and consume the proceeds from the corre- sponding investment strategies, which turns out to be dependent on the outcome of the previous date (intervention or not by the government). Figure 2.3 ilustrates the timeline of events. The asset is in total supply of K, and every agent is initially endowed with wealth A. In case the price of the asset is p, buying is affordable if and only if A≥ p. An agent that buys the asset at the second period has to liquidate it in the final one, incurring a transaction cost, t. Without buying, an agent just carries over her initial endowment to the end of the economy. All the consumption occurs at the final date, determined by the investment strategy chosen. Agents are risk neutral and the utility from consumption is equal to the proceeds from the portfolio, written as 70 t=0 t=1 t=2 t=3 State of θ,is drawn fundamentals, Investorschoose theirinvestment strategies Government observessocial welfareand whether tointervene Investors theirportfolios the proceeds liquidate andconsume Informationalscenario (imperfect,perfect orcommonprior) Framework (whethergovernment intervenesornot) Figure 2.3: Timeline of events. R i (X i ,θ,A,t)≡X i (θ−t)+A−X i p=X i (θ−p−t)+A, ∀i∈I. (2.1) If an agent chooses to buy, X i = 1, she gets the payoff from the asset minus the transaction cost to be paid, θ−t, plus whatever is left from the initial endowment after the asset was purchased, A−p. On the other hand, by not buying, X i =0, an agent keeps her total endowment, A, for later consumption. Intheframework withthepresenceofthegovernment, intervention occurswheneverso- cialwelfaregoesbelowacertainthreshold,C. Socialwelfare,S, isdefinedasthesumacross agents of theproceeds from their investment strategies, before any government intervention whatsoever, S(θ,A,t)≡ Z 1 0 [X i (θ−p−t)+A]di. (2.2) The condition for intervention is S(θ,A,t)<C (2.3) 71 and, if that is the case, the government is assumed to act 6 in such a way that results in the social welfare being restored to C. Market clearing condition, R 1 0 X i di = K, implies that, when there is government intervention, the state of affairs is equivalent to the one that would prevail in case the realization of e θ was the one implicitly given by S(θ,A,t)=C ⇔ R 1 0 [X i (θ−p−t)+A]di=C ⇔ (θ−p−t) R 1 0 X i di+A R 1 0 di=C ⇔ (θ−p−t)K +A =C ⇔ θ = C−A K +p+t≡θ ∗ . (2.4) In other words, the condition for government intervention amounts to the realization of e θ being sufficiently low, θ < θ ∗ . The interesting case is A > C for, whenever the government steps in, the resulting state of affairs is such that agents cannot fully recover the amount invested: from (2.4), conditional on government intervention, the strategy of buying the asset yields θ ∗ − p− t = (C−A)/K < 0. Not only that, if the case was A < C, investors’ problem in face of the possibility of government intervention would be lessrelevant: whateverthestrategy chosen, theresultingoutcomewouldalways bepositive. Not only that, A < C would imply government intervention even when no one invests, X i =0,∀i∈I, which is not plausible given the chosen measure of social welfare. That investors lose “money” when there is intervention is so mething that one would expect: as common sense dictates, intervention is to be associated with bad states. As such, it is not reasonable to have investors making positive profits even at those states. 6 Governmentinterventioncouldtaketheformofareductionintheinterestrateorabailoutofaparticular agent, for example. 72 Indeed, if an agent decides to invest, X i = 1, and the realized θ is one that demands government intervention, the proceeds from the investment strategy are, following (2.1), R i (1,θ ∗ ,A,t) = θ ∗ z }| { C−A K +p+t−p−t+A = C−A K +A, ∀i∈I. (2.5) Since A > C and any agent i ∈ I has the option of saving the entire wealth A for future consumption, anyone who had chosen X i = 1 instead of X i = 0 regrets having made such a choice, upon observing an intervention: while the former strategy gives utility (C−A)/K + A, the later would had delievered A; with A > C, it follows that A > (C−A)/K+A. Intervention by the government is synonymous of a bad choice when that choice was to buy the asset. Summarizing, the problem each agent faces is to build a portfolio, acknowledging that theresultingpayoffdependsnotonlyonstateoffundamentalsoftheeconomybutalsoonthe decision of the government to intervene or not, which is in turn a function of social welfare. The framework where investors are aware of the possibility of government intervention is examined in turn. 2.3 Government Intervention Intheframeworkwiththepresenceofthegovernment, investors areaware ofthepossibility ofintervention. Sincegovernment’sactionaffectsthepayoffoftheriskyasset,whenbuilding their portfolios, agents need to determine the probability of that intervention happening. Beliefs are formed in accordance with the informational scenario at hand: • Imperfect information: investors receive private signals about the realized state of 73 fundamentals, inferring the probability of government intervention from the signal obtained; • Perfect information: investors know what the state of fundamentals happens to be, immediately concluding if there will be or not intervention, according to the price at which the asset is being offered; • Common prior: all that is know is the probability distribution of the state of funda- mentals, being that the only artefact used by agents to calculate the probability of intervention. Each oftheabovescenariosisexaminedinturn,startingwiththeimperfectinformation case of the next section. 2.3.1 Imperfect Information About the Fundamentals Inthecaseofimperfectinformationaboutthestateoffundamentals,eachinvestorireceives a private signal ξ i about the realized value of e θ, denoted by θ. Signals are uniformly distributed around this realized value, e ξ ∼ U[θ−τ,θ+τ], with τ > 0. Agents know that θ is at most τ units away from the signal received, θ ∈ [ξ i −τ,ξ i +τ], ∀i ∈ I. Investors’ problem is max X i E h X i e θ−p−t +A| ξ i i =X i h E e θ| ξ i −p−t i +A (2.6) s.t. A≥p if X i =1, ∀i∈I. Agent i’s signal ξ i conveys information notonly aboutθ butalso aboutthelikelihood of government intervention. When deciding whether to buy or not the asset, each agent asks what is the probability that θ < θ ∗ , conditional on θ ∈ [ξ i −τ,ξ i +τ]. This probability in 74 turn depends on the price of the asset, since θ ∗ is a function of p: θ ∗ =(C−A)/K+p+t. For a given p, the situation a particular investor faces falls into one of three possible cases: for∀i∈I, either (I) C−A K +p+t≤ξ i −τ ; (II) ξ i −τ < C−A K +p+t≤ξ i +τ ; or (III) ξ i +τ < C−A K +p+t. In (I), since θ ∈ [ξ i −τ,ξ i +τ], it is clear that (C−A)/K + p + t ≤ θ. Therefore, conditional on the agent facing a price p and having received a signal ξ i such that (I) holds, she knows that, at the current price, there will beno government intervention. In that case the expectation of the asset payoff, e θ, is given by E e θ| ξ i = 1 2τ Z ξ i +τ ξ i −τ θdθ (2.7) = ξ i . In range (II), investors entertain the possibility of government intervention and, condi- tional on that, they calculate the expected value of e θ as E e θ| ξ i = 1 2τ " Z C−A K +p+t ξ i −τ θ ∗ dθ+ Z ξ i +τ C−A K +p+t θdθ # (2.8) = 1 2τ C−A K +p+t 1 2 C−A K +p+t −(ξ i −τ) + 1 2 (ξ i +τ) 2 . Finally, facing (III) any agent i knows the government will step in, yielding 75 E e θ| ξ i = 1 2τ Z ξ i +τ ξ i −τ θ ∗ dθ (2.9) = C−A K +p+t. Given all the threepossiblescenarios, theobjective function in theinvestors’ maximiza- tion problem, X i h E e θ| ξ i −p−t i +A, can be rewritten as X i I (I) ξ i +I (II) 1 2τ C−A K +p+t 1 2 C−A K +p+t −(ξ i −τ) + 1 2 (ξ i +τ) 2 +I (III) C−A K +p+t −p−t} +A ≡U i (X i ,ξ i ,A,p), ∀i∈I, (2.10) whereI (I) is the indicator function for case (I), namelyI (I) =1 if (C−A)/K+p+t≤ ξ i −τ and zero otherwise. Similar for cases (II) and (III). Since any investor must satisfy a budget constraint in deciding to buy, A ≥ p, and given that the entire wealth can he secured for consumption in the final period - without buying, agents are sure to get utility A - the optimal strategy is to choose X i = 1 in case the following two conditions are satisfied:        U i (1,ξ i ,A,p)≥A; A≥p. (2.11) Plugging in (2.11) the expression for U i (·) defined in (2.10) yields 76 I (I) ξ i +I (II) 1 2τ C−A K +p+t 1 2 C−A K +p+t −(ξ i −τ) + 1 2 (ξ i +τ) 2 +I (III) C−A K +p+t ≥p+t. (2.12) Hence, if it is feasible to buy the asset - A≥ p - condition (2.12) is the pivotal one for the buyingdecision. Among other things, the criterion for choosing X i =1 is dependenton p. At the equilibrium price, the fraction of agents in the interval [0,1] that satisfies (2.12) must be equal to K, the total supply. The price adjustment process - not modele d in the paper - can be thought of as a price tâtonnement process: at certain price levels demand is higher than supply, at others the converse is true: only at the equilibrium price demand equals supply,andfromaout-of-equilibriumsituation the priceevolves until anequilibrium is reached. It is argued in the sequence that an equilibrium (to be defined) can be sustained for different levels of government intervention, i.e., there are multiple equilibria associated with different levels of θ ∗ . Since the only variable that is endogenously determined in the definition of θ ∗ is p, the statement above is equivalent to saying that different price levels can be supported in equilibrium. By definition, the higher θ ∗ the higher the equilibrium p is and - as the government intervenes only if θ < θ ∗ - this implies the equilibrium price of the asset being higher the more likely government intervention is to occur. Six intervals where θ ∗ may lie can be identified, namely: (i) 0≤ C−A K +p+t≤θ−2τ ⇔0≤θ ∗ ≤θ−2τ; (ii) θ−2τ < C−A K +p+t≤θ−τ ⇔θ−2τ <θ ∗ ≤θ−τ; (iii) θ−τ < C−A K +p+t≤θ⇔θ−τ <θ ∗ ≤θ; (iv) θ < C−A K +p+t≤θ+τ ⇔θ <θ ∗ ≤θ+τ; 77 (v) θ+τ < C−A K +p+t≤θ+2τ ⇔θ+τ <θ ∗ ≤θ+2τ; and (vi) θ+2τ < C−A K +p+t≤1⇔θ+2τ <θ ∗ ≤1. The implications of θ ∗ belonging to each of the six ranges are discussed in turn. As a remark, appropriate conditions need to be imposed on the primitives of the model, in accordance with the interval where θ ∗ is to be located in equilibrium 7 . In ranges associated with a higher probability of government intervention, i.e., with a higher θ ∗ , one would expect to be easier to obtain an equilibrium - afte r all those are the cases where investing in the asset is supposedly safer. However, it is also true for those cases that the price to bepaid in order to acquire the asset, p, is higher, making the buying decision less attractive. It is the interaction of these two opposite effects that ultimately determines the overall likelihood of having an equilibrium in a particular interval. Moving from interval (i) to (vi), agents’ uncertainty changes from being related to the possibility of government absence to being related to the possibility of government intervention. From interval (i) to (iii), accurate signals indicate that the government will not intervene, whereas from (iv) to (vi), the more accurate the signal, the more one is certain that the government will intervene. Inintervals (i), (ii) and(iii), nogovernment intervention ensues, withnoagentbelieving that such an action would certainly occur. The mass of agents knowing exactly what the government behavior will be gets progressively smaller, as one goes from (i) to (iii). For instance, in (i) everyone knows the government will not intervene, in (ii) a fraction of the investors knows there will not be intervention and in (iii) relatively fewer agents can claim to be certain the government will not intervene - even though in equilibrium this is what happens. In intervals (iv), (v) and (vi), there is government intervention, with no agent being 7 The conditions for each of the six intervals are derived in a supplemental material to this paper and available upon request. 78 able to rule out such possibility. Differently from the previous intervals, however, the mass of agents who know exactly what the government behavior will be gets progressively larger, as one moves from (iv) to (vi). In (iv) some agents can claim to know for sure the government will intervene, in (v) a relatively larger fraction of the investors knows there will be intervention and in (vi) everyone knows the government will intervene, indeed. In order to make the above argument formal, recall that intervention ensues if and only if θ ∈ [0,θ ∗ ), with agents receiving signals distributed as e ξ ∼ U[θ−τ,θ+τ] and acknowledging that θ∈[ξ i −τ,ξ i +τ]. Individually, the range of uncertainty of each agent i∈ I is [ξ i −τ,ξ i +τ]. Taking the union of intervals where each investor contemplates the presence of θ yields S i [ξ i −τ,ξ i +τ] = [θ−2τ,θ+2τ], called range of group uncertainty. Following that, in interval (i) no agent belives in government intervention, since [0,θ ∗ )∩ [θ−2τ,θ+2τ] =∅: everyone is aware of the fact the government will not step in. In (ii) and (iii), on the other hand, [0,θ ∗ )∩[θ−2τ,θ+2τ]6=∅, and, as θ ∗ increases as one moves along the intervals, the fraction of agents unaware of the fact the government does not intervene becomes larger. In intervals (iv) and (v), no agent can rule out the possibility of governmentintervention: (θ ∗ ,1]∩[θ−2τ,θ+2τ]6=∅. Againusingthefactthatθ ∗ increases with the intervals, the fraction of agents aware of the fact the government does intervene becomes larger, the extreme case beinginterval (vi), where everyone knows the government will act: (θ ∗ ,1]∩[θ−2τ,θ+2τ] =∅. The degree of confusion is defined as the fraction of investors - induced by θ ∗ - which, in equilibrium, is uncertain about the behavior of the government. As a function of the critical level of government intervention, such a measure can be depicted as in Figure 2.4. An equilibrium is defined as: Definition2.1. Anequilibriumisacollection ofdecisionrules,X= n X i | X i ∈{ 0,1} ,∀i∈I o , and price p∈R ++ , such that: 79 Degree ofconfusion 1 θ ∗ 0 θ−2τ θ−τ θ θ+2τ θ+τ 1 Uncertaintyabout governmentabsence Uncertaintyabout governmentintervention (i) (ii) (iii) (iv) (v) (vi) Figure 2.4: Level of uncertainty as a function of θ ∗ . (i) given price p and private signal ξ i , X i ∈argmax{ U i (X i ,ξ i ,A,p)|A≥p if X i =1} , ∀i∈I; and (ii) market clears: R 1 0 X i di=K. For θ ∗ lying in any of the six intervals previously listed, the method used to obtain an equilibrium is the following: (i) Postulate that θ ∗ lies in a particular interval among those from (i)-(vi); (ii) Using the market clearing condition, determine the equilibrium price; (iii) Impose conditions on the primitives of the model such that, at the equilibrium p, θ ∗ lies in the interval postulated in the first step and the conditions for an equilibrium given in Definition 2.1 are satisfied. The aforementioned method is illustrated in Figure 2.5. Deriving the conditions such that the critical level of government intervention, θ ∗ , can be located in any of the six inter- vals, two factors turn out to be important - as far as the funct ional form of the equilibrium 80 p t Agentsobserveθ ∗ and decidewhethertobuyornot, X i ∈{ 0,1} , ∀i∈I R I X i d i <K R I X i d i =K R I X i d i >K (3) p t ↓p t+1 p t+1 =p t p t ↑p t+1 (4) Returnto (1) Equilibrium is reached Returnto (1) (5) (1) (2) Figure 2.5: Equilibrium process. price and the corresponding comparative statics are concerned: (i) The magnitude of τ relative to the downside risk (A−C)/K; (ii) The supply level of the asset, K. Regarding τ, the high (low) uncertainty case is the one where τ > (<)(A−C)/K. Recall that τ is the parameter definingthe lenght of the range of uncertainty of each agent, and also the support of the distribution of signals: a higher τ means both that investors contemplate a larger interval where the realized θ might be located and that the signals are more spread out. AboutK,thesupplylevel, twocasesaredistinguished: oneof low supply,with0<K ≤ 1/2, and one of high supply, where 1/2 < K ≤ 1. Given the low/high dichotomy in terms of uncertainty, τ, and the supply level, K, the asset and its correspondingly equilibrium price can be characterized in two dimensions, given in Table 2.1. Table2.1presentsthefunctionalformoftheequilibriumprice,foreachpossiblecombina- 81 Uncertainty level, τ Supply level, K τ < (A−C)/K τ > (A−C)/K 0<K ≤ 1/2 P =θ+2τ (1−K)+(A−C)/K− 2[τ (A−C)/K] 1/2 +t P =θ+τ (1−2K)−t 1/2 <K ≤ 1 P =θ+2τ (1−K)+(A−C)/K− 2[τ (A−C)/K] 1/2 +t P =θ+τ (1−2K)−t Table 2.1: Asset price characterization in terms of uncertainty and supply level. tion ofuncertainty andsupplylevel. Thecomparative statics follow fromthecorresponding expressions. Two results emerge: (i) For a given level of uncertainty, the equilibrium price (functional form) is uniform across assets in low and high supply; (ii) For a given level of supply, the comparative statics are uniform across assets of low and high uncertainty. Figure 2.6 depicts the the equilibrium price and comparative statics according to the downside-risk intervals 8 , uncertainty and supply level. If the critical level of government intervention, θ ∗ , is to lie in a particular range from (i)-(vi), the correspon ding downside- risk (A−C)/K has to be within the appropriate interval in the figure. The first line corresponds to the low supply case, measuring the downside risk: if (A−C)/K is greater (smaller) than τ, theasset is oneof low (high)uncertainty. Analogously for thesecond line, which corresponds to the case of high supply. Regardlessoftheuncertaintyandsupplylevel,thecomparativestaticsoftheequilibrium price, P, with respect to the state of fundamentals θ, wealth A, insurance C, supply K, and transaction cost t, are the same, namely: (i) P θ >0; (ii) P A <0; (iii) P C >0; 8 These intervals are also derived in the supplemental material to the paper. 82 P=θ+τ(1-2K)-t P=θ+2τ(1-K)+(A-C)/K -2[ τ(A-C)/K] 1/2 -t 0 τ[(1-2K)/2] 2 τ(1-K) 2 τ 2τ(1-K) τ(3-2K) (v) (iv) (iii) (iii) (ii) (i) 0 τ τ(1-K) 2 4τ(1-K) 2 τ[(3-2K)/2] 2 2τ(1-K) τ(3-2K) (iv) (ii) (A-C)/K (i) (iii) (iii) (ii) (A-C)/K -Low supply -High uncertainty -P τ>0 -Low supply -Lowuncertainty -P τ>0 -High supply -High uncertainty -P τ>0 -High supply -High uncertainty -P τ<0 -High supply -Lowuncertainty -P τ<0 Figure 2.6: Downside-risk intervals and equilibrium price according to uncertainty and supply levels. (iv) P K <0; (v) P t <0. Asonewouldexpect,theequilibriumpriceisincreasinginboththestateoffundamentals and the insurance provided by the government - the safety net - whereas it is decreasing in the supply level and the transaction cost. Regarding the comparative statics with respect to wealth, the reason of the price decreasing is the enhanced attractiveness of the saving for future consumption option investors have: by not buying, a consumption level on par with the initial wealth can be secured. The more investors save, the more the demand 83 falls, depressing the equilibrium price of the asset. Another way of seeing it, recall that (A−C)/K is the downside risk, i.e., the loss that buying the asset would entail when θ <θ ∗ ,vis-a-vistheoptionofsavingthewealthforfutureconsum ption. IncreasingAmeans the downside risk getting larger, and buying the asset riskier relative to not participating in the market 9 . Regarding the comparative statics with respect to a broadening of the range of uncer- tainty, the effect on the equilibrium price of an increase in τ depends only on the supply level of the asset - the reason why it was previously stated th at, for a given level of supply, thecomparative statics areuniformacross assets of low and highuncertainty. Iftheasset is in low supply, the effect on theprice of an increase in τ is positive, whereas if theasset is in high supply, thateffect is negative. All inall, government intervention makes an increasein the level of uncertainty to generate extra potential gains without the commensurate effect on the side of potential losses - after all, investors know th e state of affairs will be never below a certain level, which by definition is the critical level of government intervention. Alone, this effect would undoubtedly lead to a price increase. There is, however, a second effect to be considered resulting from an increase in the rangeofuncertainty. For, recall from(2.5) that, wheneverthereisgovernment intervention, any investor i ∈ I would had been better-off had she saved her entire wealth for f uture consumption, X i = 0, rather than choosing to buy. Therefore, changes that lead to an increaseinthelikelihoodofgovernmentinterventionrelativetothelikelihoodofgovernment absence make buying the asset, X i =1, less of an attractive choice. Tomaketheargumentformal, noticefromFigure2.7that, foranagenti∈I withsignal ξ i , the probability of government intervention is given by 9 Just to reinforce the point, agents are heterogenous only with respect to the information received, which implies that the decision of not buying results from having a low signal, not from lacking resources. Therefore, increasing the wealth of agents will not induce more agents to participate in the market, which would possibly increase demand and, consequently, the price. 84 Figure 2.7: Support of e θ, distribution of signals, range uncertainty and location of θ ∗ . Pr(Intervention| ξ i )= θ ∗ −(ξ i −τ) 2τ , (2.13) whereas the probability of government absence is Pr(Absence| ξ i )= (ξ i +τ)−θ ∗ 2τ . (2.14) Defining a function Δ of the difference between the two, namely 85 Δ i (θ ∗ ,τ;ξ i ) ≡ θ ∗ −(ξ i −τ) 2τ − (ξ i +τ)−θ ∗ 2τ = θ ∗ −ξ i τ , ∀i∈I, (2.15) and taking the derivative with respect to τ yields ∂ ∂τ Δ(θ ∗ ,τ;ξ i ) = ξ i −θ ∗ τ 2 =        >0 if ξ i >θ ∗ , <0 if ξ i <θ ∗ . (2.16) Since the critical level of government intervention, θ ∗ , is decreasing in the supply level, K, the additional probability of intervention that results from an increase in the range of uncertainty - ∂Δ/∂τ above - is higher when the supply is high than when it is low. Th e intuition is that, the more the asset permeates the economy - the case of high supply - the more social welfare is exposed to changes in τ. Since social welfare is the measure upon which the intervention decision is taken, the likelihood of the government stepping in end us increasing with the range of uncertainty. It is the interaction of these two effects that determines the overall impact of changes in τ: first, the benefit of having an extra exposure to potential gains without having to incur the risk of additional losses, second the increase in the probability of government intervention that makes buying less of an attractive choice relative to saving the wealth for future consumption. In the low supply case, the first effect is more important, as the downside risk per unit of the asset, (A−C)/K, is large. In the case of high supply, the risk is spread among a larger number of investors, with an increase in uncertainty not 86 adding much to the prospects of buying the asset - in fact, mor e uncertainty ends up being detrimental. This is the reason why the equilibrium price increases with uncertainty, if the asset is in low suppy, and decreases otherwise, i.e., if the supply happens to be high. Making a parallel between supply and liquidity level, it could be said that assets in low supply would correspond to assets of low liquidity, whereas assets in high supply would be ones of high liquidity. From this perspective, the above result could be translated into saying that an increase in the range of uncertainty is beneficial for investors long in assets of low liquidity and detrimental for those invested in assets of high liquidity: as it was just argued, the price of the asset increases with uncertaity in the firstcase and decreases in the second. From a self-selection point of view, that would be re asonable: investors who opt for assets of low liquidity reveal their preferences for risky investments and, accordingly, an increase in the level of uncertainty would beseen positively. Those who choose investments of high liquidity tend to choose safer investments, making an increase in uncertainty less attractive 10 . Returning to Figure 2.1, a close inspection shows that the functional form of the equi- librium price is the same across assets in low and high supply - conditional on the level of uncertainty. As it turns out, the equilibria in the low uncertainty case involve no govern- ment intervention. Shown in Figure 2.6, the equilibrium price in this case does not depend on the safety net provided by the government, C, leading one to believe that, after all, the possibility of intervention is not important if the asset’s payoff is characterized by low uncertainty. This is misleading, however: among those intervals where in equilibrium there is no intervention, i.e., (i)-(iii), only in the first one - th e one where the critical level of government intervention is at its lowest - the signals recei ved by the agents are fully infor- mative of the government’s behavior. For the others, even though some investors still know 10 This argument is not pushed further since agents are risk neutral and, therefore, are not supposed to have preferences towards more or less risk. This could be explored were the agents endowed with different preferences. 87 the government safety net will be absent, some others cannot reach the same conclusion, which affects the way they calculate the expected payoff of buying the asset and, therefore, their investment decision. With high uncertainty, the equilibrium price is clearly dependent on the level of in- surance, as Figure 2.6 indicates. Differently from the low uncertainty case, however, the government safety net appears in the equilibrium price not only in those intervals where there is intervention but also in those where the government ends up not stepping in. A high level of uncertainty seems to be crucial if changes in the safety net, C, are to have an impact in the equilibrium price. Accordingly, one could well argue that the possibility of intervention has more of an impact in turbulent environments - those characterized by a high level of uncertainty. As discussed previously, however, this is not to say that the possibility of government intervention is unimportant in situations of low uncertainty, it is just that - with high uncertainty - the presence of the govern ment is important enough to leave its footprints in the equilibrium price. As one can deduce from Figure 2.6, an equilibrium cannot be supported in interval (vi), where the critical level of government intervention, θ ∗ , is at its highest. The reason is that, with such a θ ∗ , agents infer they are paying a high price to invest, after all insurance provided by the government is costly. A high price makes the strategy of buying more likely to perform badly: only if the asset performs extremely well can the investment be salvaged. Facing ahigh price, some investors prefertoabstain frombuyingtheasset, which in extreme can lead to the market clearing condition not being satisfied. Unless the price decreases sufficiently enough, no equilibrium can be supported, as in the process depicted in Figure 2.5. This non-existence of equilibrium could be related to the bu rst of a bubble, indicating theimpossibilityofpricesgoingbeyondacertainlevel. Justaswhenabubbleburststhereis not enough demand to keep prices increasing, so is the non-ex istence of equilibrium caused 88 by a lack of investors willing to buy the asset, as few want to commit to such a costly investment - implicitly in those instances where the probab ility of intervention is high 11 . 2.3.2 Perfect Information The perfect information case is trivial but mentioned here, nonetheless. In this case, in- vestors know the state of fundamentals, θ, so that the critical level of government interven- tion, θ ∗ , leads to one of two possible outcomes: (i) 0≤ C−A K +p+t≤θ⇔ 0≤θ ∗ ≤θ - no government intervention; (ii) θ < C−A K +p+t≤1⇔θ <θ ∗ ≤1 - government intervention. Investors’ maximization problem is max X i X i (θ−p−t)+A (2.17) s.t. A≥p if X i =1. Therefore, in case agents can afford to buy the asset, A≥p, they choose X i =1 only if θ−p−t+A≥A⇔θ≥p+t. If that is the case, A>C implies that [p+t,1]∩[(C−A)/K +p+t,1] =[p+t,1], (2.18) whereas [p+t,1]∩[0,(C−A)/K +p+t)=∅, (2.19) 11 As themodel is static and a specific definition of a bubbleis not provided, this remark is just to indicate a point that could be further explored in an extension of the paper. 89 whichmeansanequilibriumcanbesupportedonlyintherangewherethereisnogovern- ment intervention. Hence, defining p PI as the equilibrium price in the perfect information case, in equilibrium θ ≥ p PI +t ⇔ p PI ≤ θ−t. With the assumption that 0 ≤ K ≤ 1, i.e., the supply of the asset is never greater than potential demand, competition would bid the price up, to the point where p PI = θ−t, which would constitute the maximum price investors would be willing to pay knowing the government will not act. Followingtheexpressionoftheequilibriumprice,thecomparativestaticsarestraightfor- ward: the price is increasing in the state of fundamentals and decreasing in the transaction cost, as one would expect. Thereis no downsiderisk to beconsidered by the investors since they know what the payoff of the asset will be. Because of that, the wealth level A, gov- ernment insurance C and supply K do not appear in the equilibrium price, not mattering for the comparative statics. With perfect information, either all investors choose to buy or everyone refrains from doing so - the heterogeneity in information from the previou s scenario is gone. In case all investors choose to buy the asset, if the supply is less than one - i.e., supply is less than demand - some mechanism must be available so that only a f raction of the agents get selected to make the investment. As discussed before, if the wealth constraint is not binding, those who are not selected would be willing to make a higher offer in order to participate in the market, which would increase the price up to the point where no one is willing to pay a price higher than the one prevailing. In this sense, there is a combined effect on the equilibrium price coming from the interaction of the supply and wealth levels, effect that is absent if one is to look only to the expression of the equilibrium price. 2.3.3 No Information: Common Prior About the Fundamentals The common prior scenario refers to the case where agents have no information other than the distribution of e θ, which is assumed to be common knowledge and given by e θ∼U[0,1]. 90 Investors’ problem is max X i E h X i e θ−p−t +A i =X i h E e θ −p−t i +A (2.20) s.t. A≥p if X i =1, and, as usual, they acknowledge that intervention is in place only if θ <θ ∗ , a condition whichis equivalent to θ <(C−A)/K+p+t. Also, agents know thatgovernment interven- tion makes the prevailing situation to be equivalent to one where the state of fundamentals equals θ =θ ∗ =(C−A)/K +p+t. Hence, E e θ = Z 1 0 θdθ = Z C−A K +p+t 0 θ ∗ dθ+ Z 1 C−A K +p+t θdθ = 1 2 " 1+ C−A K +p+t 2 # . (2.21) In case agents can afford to buy the asset, A≥p, they choose X i =1 only if 1 2 " 1+ C−A K +p+t 2 # −p−t+A≥A ⇔ 1 2 " 1+ C−A K +p+t 2 # −p−t≥0. (2.22) The left-hand side of the above inequality has two roots, nam ely 91 p ′ = A−C K −t+1− s 2 A−C K , (2.23) p ′′ = A−C K −t+1+ s 2 A−C K . (2.24) For any p≤A such that either p≤ p ′ or p ′′ ≤ p, condition (2.22) is satisfied. With the assumption that 0≤ K ≤1 - supply is never greater than potential demand - competiti on wouldbidthepriceup. Denotingp CP astheequilibriumpriceinthecommonpriorscenario, the following outcomes might emerge, according to the wealth level A: (i) 0<A≤p ′ ⇒p CP =A; (ii) 0<p ′ <A<p ′′ ⇒p CP =p ′ ; (iii) 0<p ′ <p ′′ ≤A⇒p CP =A. If supply is strictly less than potential demand, 0≤ K < 1, the agents allowed to buy the asset get selected randomly from the population of investors 12 . Notice that, if p CP =A, θ ∗ =(C−A)/K+A+t and, since θ ∗ ∈[0,1], it must be that (A−C)/K < A+t. Analogously, if p CP = p ′ , a feasible θ ∗ requires (A−C)/K < 1/2. Therefore, an equilibrium with the possibility of government intervention - i.e., one with a feasible θ ∗ - can be supported only if the downside risk, ( A−C)/K, is not too high or, in other words, if the support from the government in a crisis is sufficiently large 13 . From the possible outcomes specified above, a wealth level A either sufficiently small 12 As in the perfect information framework, there is no heterogeneity whatsoever among the agents when all they know is the distribution of e θ, implying the equilibrium to be one where either none or everyone is willing to buy the asset. Since the mass of investors is equal to one, if the supply K is less than that there must exist a device such that only a fraction of the investors will be selected to buy - in case them all want to do so. 13 Otherwise, i.e., if θ ∗ < 0, every agent believes that the government will be absent, since it is known that intervention happens only if θ < θ ∗ and θ ∈ [0,1]. That being the case would imply E e θ = 1/2, automatically changing the analysis above. 92 or sufficiently high implies the equilibrium price being p CP =A, with corresponding trivial comparative statics. With an intermediate wealth level, it follows that p CP = A−C K −t+1− s 2 A−C K ≡P (K,t,A,C). (2.25) With the condition (A−C)/K ≤1/2 in place, the comparative statics are: (i) P K = 1 K q (A−C)/K 2 − A−C K >0; (ii) P t =−1<0; (iii) P A = 1 K 1− 1 √ 2(A−C)/K <0; and (iv) P C = 1 K 1 √ 2(A−C)/K −1 >0. With respect to K, there are two effects to be considered. The first one is the usual positive supply shock - price decrease. The second - and, as i t turns out, the relevant one - is the decrease in the individual downside risk. The downside risk associated with a change in social welfare upon an intervention is still the same - A−C - but that risk per investor - (A−C)/K - decreases when K increases. Individually, therefore, each investor bears less risk when the supplyof the asset goes up. That makes the buyingdecision more attractive, increasing the demand and, consequently, the price 14 . Regarding A, one must bear in mind that the comparative statics at hand hold for a wealth level p ′ < A < p ′′ . Since paying any price p ∈ (p ′ ,p ′′ ) yields a negative expected utility, amarginalincreaseinwealthwillnotincreasethedemandfortheasset. Nonetheless, there is a secondary effect that comes from changes in A that does affect the price: an increaseinwealthmakestheoptionofsavingforfutureconsumptionmoreattractiverelative 14 This effect can be seen as a negative externality investors cause on each other: from their own perspec- tives, buying the asset becomes more attractive since less risky, but that in turn increases demand and, ultimately, the price everyone else has to pay upon deciding to invest. 93 to buying, inducingagents to demand a pricecut if they are expected to invest in the asset. This is what ultimately happens and is reflected in P A <0. Finally, the comparative statics with respect to t and C are as expected: an increase in the transaction cost decreases the price of the asset whereas if the insurance offered by the government increases - C goes up - so does the equilibrium price. 2.3.4 Analysis of the Equilibrium Price Across Different Informational Scenarios After deriving the expression of the equilibrium price - and respective comparative statics - for each informational scenario, the focus now is on the com parison of those prices - the intra framework, inter scenario analysis of Figure 2.2. The question one asks is how the interaction of the degree of information with the possibility of intervention affects the equilibrium outcome. Recall first that, in the imperfect information scenario, the magnitude of τ with respect tothedownsiderisk-i.e., thelevelofuncertaintyeachage ntfaceswhenmakingthedecision to invest - was one of the determinants of the equilibrium pri ce. Second, for the perfect information and common prior scenarios, in equilibrium either all or none of the investors choose to buy the asset. Following suit, our analysis henceforth will distinguish between scenarios of low and high uncertainty, for each of them the supply level is assumed to be unitary. With unitary supply, for the low uncertainty case the equilibrium prices are: (i) Imperfect information: p IP =θ−τ −t; (ii) Perfect information: p PI =θ−t; (iii) Common prior 15 : p CP =A or p CP =A−C−t+1− p 2(A−C). 15 In the common prior case, p CP =A if the wealth constraint is binding. 94 Since τ >0, it is clear that p PI >p IP . Also, as the asset is affordable to investors only if wealth is higher than price, regardless of the state of fundamentals the ordering of prices is p CP >p PI >p IP , in case p CP =A. More interesting is the case where the wealth constraint is not binding and the price in the common prior framework is given by p CP = A−C−t+1− p 2(A−C). Following expression (2.25), it was pointed out that, in order to be consistent with the possibility of government intervention, the downside risk must be such that 0≤A−C ≤1/2. With this assumption in place, it is shown in the appendix - Propositio n B.1 - that, for a sufficiently high realization of θ, the ordering of equilibrium prices is p PI >p IP >p CP , (2.26) whereas for an intermediate level, p PI >p CP >p IP , (2.27) with the following prevailing if the state of fundamentals happens to be sufficiently low, p CP >p PI >p IP . (2.28) Figure 2.8 summarizes the ordering of equilibrium prices for the low uncertainty case, according to the state of fundamentals. For any level of θ, the equilibrium price with perfect information, p PI , is higher than the one that prevails in the imperfect information framework, p IP , reflecting the cost of information - more information comma nds a higher price. Ifinvestorsonlyhaveknowledgeofthedistributionof e θ -thecommonpriorscenario- theexpectedstateoffundamentalstheyenvisageishigherthanθwhenthatrealizationislow andlower thanthatwhenthatrealization ishigh, whichisthereasonwhyp CP >p PI >p IP 95 for a sufficiently low θ and p PI >p IP >p CP for a sufficiently high. Figure 2.8: Equilibrium price ordering across different informational scenarios for the low uncertainty case. Turningtothehighuncertaintycase,theequilibriumpricesacrossthedifferentscenarios, with a unitary supply, are: (i) Imperfect information: p IP =θ+A−C−t−2 p τ(A−C); (ii) Perfect information: p PI =θ−t; (iii) Common prior: p CP =A or p CP =A−C−t+1− p 2(A−C). As Proposition B.2 intheappendixshows, all inall theorderingofequilibriumprices in the high uncertainty scenario is similar to the one obtained for the low uncertainty one. As before, the equilibrium price when investors have perfect information is always higher than theone prevailing whenall they observe is a private signal aboutthestate of fundamentals. Thepricedifferentialis,again,justaconsequenceofthecostofinformation: better(perfect) information commands a higher price relative to worse (imperfect) information. Regarding the equilibrium price in the common prior scenario, for a sufficiently low realization ofthestateoffundamentals, theunconditionalexpectation of e θ is higherrelative to that in the other scenarios, and so is the equilibrium price. By the same token, that unconditional expectation is lower relative to the others for a sufficiently high level of fundamentals, and so lower is the equilibrium price. Figure 2.9 summarizes the ordering of prices for the high uncertainty case 16 . 16 As Proposition B.2 in the appendix shows, τ being greater or less than 1/2 turns to be important for 96 Figure 2.9: Equilibrium price ordering across different informational scenarios for the high uncertainty case. 2.4 No Government Intervention Inthesetupwithnogovernment, investorsacknowledgetheimpossibilityofanintervention, regardless of the state of fundamentals coming to the fore: by assumption, that is ruled out from the outset. The insurance or safety net they would otherwise rely on, as in the framework of the previous section, is now nonexistent. Agents therefore must take this new paradigm into account when calculating the expected payoff of buying the asset. The informational constraints, however, are still there: the scenario might still be one of imperfect, perfect or common prior information, in the same fashion of the previous analysis. The imperfect information case of the next section is discussed first. the ordering of equilibrium prices in the high uncertainty scenario: the ordering in the first line of Figure 2.9 corresponds to the case where τ < 1/2, whereas in the third it is considered τ > 1/2. Being simply a technicality, this point is ignored in the text. 97 2.4.1 Imperfect Information About the Fundamentals Withnogovernmentandinvestorsreceivingaprivatesignalaboutthestateoffundamentals -bydefinition,theimperfectinformationscenario-onekno wsfrom(2.7)that, foranyi∈I, E e θ| ξ i = ξ i , and, from (2.12), that agents opt to buy only if ξ i ≥ p+t - in case buying the asset is affordable, A≥p. The market clearing condition is 1 2τ Z θ+τ p+t dξ =K ⇔ θ+τ−p−t=2τK ⇔ p=θ+τ(1−2K)−t≡p NG . Proposition B.3 in the appendix establishes the necessary conditions under which p NG can be supported as an equilibrium price. The comparative statics are derived from the expression of p NG . In that regard, θ, K and t are related to the equilibrium price as one would expect: p NG is increasing in the state of fundamentals and decreasing in both the supply level and the transaction cost. Similarly to the comparative statics for the imperfect information scenario with the possibility of intervention, here the effect of changes in τ - the parameter that commands therange of uncertainty each agent faces -dependson thesup plylevel of theasset, K. For, itis clear from theexpressionofp NG that thepriceincreases in τ ifK <1/2, anddecreases otherwise. With risk-neutralagents, onewouldatfirstbelievethatcha ngingτ wouldnotcauseany effect on the equilibrium price - after all, with such prefere nces, investors’ concerns regard only the expected value of e θ, and an increase in the range of uncertainty would constitute nothing more than a mean-preserving spread . However, the market clearing condition - being a restrictio n on the proportion of in- vestors buying the asset - makes the supply level pivotal in d etermining the effect on the 98 equilibriumpriceofchangesinτ. Themarketclearingconditionimpliesthattheproportion of investors buying the asset is θ+τ−p−t 2τ =K. (2.29) Changing τ has an effect on both the numerator and the denominator of the ratio in the left-hand side of relation (2.29), while the right-hand side is fixed. If K >1/2, changes in the numerator must be overall greater than those in the denominator. Since the price enters theexpressionwith anegative sign, adecrease init mustfollow, leadingto P NG τ <0. With alow supplylevel, K <1/2, theconverse appliesand, therefore, theequilibriumprice increases with τ. As in the framework with the possibility of government intervention, if an analogy between supply level and liquidity is to be made, the above could be translated as the following: an increase in the uncertainty level of assets in low supply has a positive effect on prices, whereas for assets in low supply that effect is negative. As argued before, from a self-selection point of view this would be reasonable: tho se investing in assets of low liquidity-byrevealingapreferencetowardsrisk-wouldap preciateanincreaseinτ, causing a corresponding increase in the equilibrium price, the opposite being the case for those investing in assets of high liquidity. Since investors are risk-neutral and, therefore, do not exhibit any preference for more or less risk, this point is not elaborated further. 2.4.2 Perfect Information About the Fundamentals Theperfectinformation scenario, as in the framework with the presenceof the government, is trivial but mentioned nonetheless. The investors’ maximization problem is 99 max X i X i (θ−p−t)+A (2.30) s.t. A≥p if X i =1. Therefore, in case agents can afford to buy the asset, A≥p, they choose X i =1 only if θ−p−t+A≥A ⇔ θ≥p+t. (2.31) Denoting the equilibrium price by p NGP , it is the case that θ≥p NGP +t ⇔ p NGP ≤θ−t. (2.32) For a supply level as large as potential demand 17 , 0<K ≤1 , competition for the asset bids the price up, to the point where p NGP = θ−t: this constitutes the maximum price investors would be willing to pay given that government will not intervene. From p NGP = θ−t, the comparative statics are trivial: the price is increasing in the state of fundamentals and decreasing in the transaction cost. There is no effect from the government safety net and uncertainty: first, by assumption the government is absent and, second, there is no uncertainty, given that the realization of e θ is known - it is the perfect information scenario after all. 17 As remarked in other instances, if supply is strictly less than demand, 0 < K < 1, it is assumed that investors are selected randomly to buy the asset. 100 2.4.3 No Information: Common Prior About the Fundamentals Finally, the focus turns to the case where agents do not receive any signal, being aware only of the distribution of e θ, given by e θ ∼U[0,1] and assumed to be common knowledge. Investors’ problem is max X i E h X i e θ−p−t +A i =X i h E e θ −p−t i +A (2.33) s.t. A≥p if X i =1. Taking into account the impossibility of intervention, it follows that E e θ = Z 1 0 θdθ = 1 2 . (2.34) Therefore, in case agents can afford to buy the asset, A≥p, they choose X i =1 only if 1 2 −p−t+A≥A ⇔ 1 2 −p−t≥0. (2.35) The left-hand side of the above inequality has one root, name ly p ′ = 1 2 −t. For any p ≤ A such that p ≤ p ′ , condition (2.35) is satisfied. If, as in the perfect information case, it is assumed that 0 < K ≤ 1, i.e., supply is never greater than poten- 101 tial demand, competition forces the price to its maximum. Hence, denoting by p NGC the equilibrium price, the possible outcomes are (i) 0<A≤p ′ ⇒p NGC =A; (ii) 0<p ′ <A⇒p NGC =p ′ . With the equilibrium price being either p NGC =A or p NGC =1/2−t, the comparative statics are trivial. For the first case, where the wealth constraint is binding - in the sense that it is consumed in its entirety when investors buy the asset - an increase in wealth leads clearly to an increase in the equilibrium price. For the second, all that matters is the expected valueof the state of fundamentals -1 /2, which is fixed- andthetrasaction cost to be paid in the final period. In this way, only changes in t have an effect on the equilibrium price - in particular, a negative one. 2.4.4 Analysis Withequilibriumpriceexpressionsforthedifferentscenarios, namelyp NG ,p NGP andp NGC -imperfect,perfectandcommonpriorinformation,respect ively-acomparisonacrossprices can be made: the so called intra framework, inter scenario analysis of Figure 2.2, in the same fashion as the one done in the framework with the presence of the government. The objective is to better understand how the complete absence of government supportinteract with different levels of payoff uncertainty in determining equilibrium prices. Tobeginwith,sincewithperfectandcommonpriorinformationtheequilibriumoutcome is such that either none or all investors buy the asset, it is assumed henceforth a unitary supply level. With that in place, the expressions of equilibrium prices are: (i) Imperfect information: p NG =θ−τ−t; (ii) Perfect information: p NGP =θ−t; 102 (iii) Common prior: p NGC =A or p NGC =1/2−t. If the wealth constraint binds in the common prior case - impl ying p NGC = A - it follows that p NGC > max n p NG ,p NGP o , since affordability requires the price of the asset being smaller than wealth. That τ > 0 in turn implies p NGP > p NG and, accordingly, the ordering of prices is p NGC >p NGP >p NG . The wealth constraint not binding in turn implies p NGC = 1/2−t and, in this case, Proposition B.4 in the appendix shows that, for a sufficiently low realization of the state of fundamentals, 0≤θ <1/2, the ordering of equilibrium prices is p NGC >p NGP >p NG , (2.36) whereas for a sufficiently high, 1/2<θ≤1, the ordering is given first by p NGP >p NG >p NGC , (2.37) - if 1 /2+τ <θ≤1 - and second by p NGP >p NGC >p NG , (2.38) - in case 1 /2 <θ <1/2+τ. Figure 2.10 summarizes the orderings given above. Regardless of the realization of e θ, it is always the case that p NGP > p NG , i.e., the equilibrium price in the perfect information scenarioisalwayshigherthantheoneprevailingwheninvestorshaveimperfectinformation: better (perfect) information, after all, commands a higher price than worse (imperfect) information. For a sufficiently low realization of the state of fundamentals, 0 ≤ θ < 1/2, the un- conditional expected value assigned by investors to the asset’s payoff, 1/2, is higher than 103 the value investors would expect in both the perfect and imperfect information scenarios, and accordingly higher is the price they accept to pay, p NGC , compared to what would be agreeable in the other scenarios. This argument is reversed for sufficiently high state of fundamentals, in particular for 1/2+τ < θ < 1, in which case p NGC is the lowest of the equilibrium prices. Figure 2.10: Equilibriumprice orderingacross different informational scenarios withoutthe possibility of government intervention. 2.5 Government vs No Government Prices Fromtheanalysisinthetwoframeworks- withand withouttheparticipation ofthegovern- ment - across the different informational scenarios - imperfect, perfect and common prior information - the conclusion is that, overall, prices respect the follow ing order: • For a high realization of the state of fundamentals, the perfect information price dominates, whereas the common prior price is dominated; • For a moderate realization of the state of fundamentals, the perfect information price still dominates, whereas the imperfect information price is the one dominated; • For a low realization of the state of fundamentals, the common prior price dominates, while the imperfect information price is dominated. This conclusion can be drawn from a comparison of Figure 2.8, 2.9 and 2.10. The possibility of intervention, therefore, does not seem to alter significantly the ordering of 104 equilibrium prices when a comparison is made across the different informational scenarios. The possibility of intervention reveals to be more substantial when one performs the inter framework, intra scenario analysis of Figure 2.2, which is done in turn. 2.5.1 Imperfect Information Intheimperfectinformationscenario,comparingtheequilibriumoutcomethatemergeswith the possibility of government intervention to the one that prevails when the government is absent, it follows that the later can be viewed as a special case of the former, the special case being the one where the critical level of government intervention is so low that - no matter what the state of fundamentals happens to be - every in vestor knows there will not be any intervention. Recalling that the higher the critical level of government intervention, the higher the equilibrium price, the above translates into saying that only at those prices deemed to be lowest in the framework with government participation can an equilibrium be supported in a framework where that very same government is absent. In other words, all the equilibria with higher prices that could be supported were the agents to entertain the possibility of an intervention cease to exist when that intervention is ruled out from the outset. It is precisely the possibility of government intervention, therefore, that allows higher prices to emerge an be supported as equilibria 18 . 2.5.2 Perfect Information In the perfect information scenario, the equilibrium price is the same in both frameworks, with and without the participation of the government. Recall that, in the setup where the government is allowed to participate, an equilibrium can be sustained only when investors 18 That higher prices can emerge in the presence of the government but not otherwise could be interpreted as the case of the government inducing a bubble. This path is not pursued, however, since what is behind those higher prices is a factor which, if not related intrinsically to the fundamentals of the asset, still affects the investors’ payoff, nevertheless: that is the insurance p rovided by the government. 105 pay a price that is conducive to an outcome wherethe government ends up not intervening. In others words, the only type of equilibrium in the perfect information scenario, with the participation of the government, is the one where the government does not intervene - and this is the reason why the equilibrium price is identical to that in the setup without the possibility of intervention. The reason why an equilibrium cannot be supported when government intervention is certain is the following. Since the government steps in only when the realized state of fundamentalsfalls belowacritical level-withthatcritical level beinganincreasi ngfunction of the price of the asset - when investors are certain of gover nment intervention, odds are thatthecriticallevelisbeingsettoohighand,accordingly, toohighisthepricebeingasked. Another way of seeing it, when investors pay too high of a price, the level of fundamentals required to make the investment profitable is correspondingly high, which makes it more likely that investors will suffer losses and, hence, more likely that the government will intervene, which is just what a high critical level implies. The price being too high pushes investors away frominvesting intheasset, causingafailureof themarket clearingcondition and preventing an equilibrium to emerge. 2.5.3 Common Prior When investors have only knowledge of the distribution of the state of fundamentals - the common prior scenario - the equilibrium price in the framewo rk with the possibility of government intervention is at least as high as theone that results inthe framework without the participation of the government. Proposition B.5 in the appendix shows this result formally. The reason is straightforward: since investors do not receive any information regarding the level of fundamentals, all they calculate is the unconditional expectation of e θ. In the framework without the presence of the government, that expectation is taken over the 106 whole supportof the distribution of e θ, whereas in the alternative framework, the supportis truncatedatthecriticallevelofintervention-afterall, investorsknowthepayoffoftheasset will not be lower than the critical level. This truncation makes the expected value of the state of fundamentals higher in the framework with the government than in the framework without, with the same conclusion being carried to the equilibrium prices. 2.5.4 Informational Scenarios Combined: the Main Result Combining the conclusions above - obtained after a comparis on across frameworks of equi- librium prices under each informational scenario - yields t he main result, stated in the following Theorem 2.1. Regardless of the informational scenario faced by investors being one of imperfect, perfect, or common prior information, the resulting equilibrium price is at least as high in the framework with the presence of the government relative to the one where the government’s safety net is absent. 2.6 Concluding Remarks The paper studies the effects on prices of the possibility of government intervention in situations of distress. This is contrasted with a framework where investors acknowledge the nonexistence of a safety net that could otherwise attenuate potential losses. In order to understand the consequences in a diverse class of investments, different informational scenarios are used as proxies for the heterogeneity in assets’ characteristics. In terms of modelling assumptions, first, investors are taken to be risk neutral, allowing one to concentrate on those instances where the possibility of intervention and the degree of information are the only drivers behindagents’ decision, rather than preferences towards lessormoreriskthatcouldadventfromhavingtheutility functionspecifiedinanotherway. 107 Second, upon deciding to invest, agents can only buy a single unitof a risky asset, with the leftovers from the wealth they are all endowed with being saved for future consumption, as if invested in a riskless asset paying no interest. This turns the investment decision into a choice of whether or not to participate in a market - here bein g called stock market - which can beas importanta factor of impact on prices as thetradingactivities of a large investor. Upon deriving the equilibrium price for each combination of framework and scenario - i.e., a choice of setup with or without government’s presence, the information agents are endowedwithbeingeitherimperfect,perfectorcommonprior-itemergesthatgovernment’s presence does not affect the ordering of prices when a comparison is made across scenarios: invariably, according to the state of fundamentals, the relation among prices derived under the assumption of imperfect, perfect amd common prior information is the same. However -andthis is oneof themain resultsobtained -theequ ilibriumpricesunderthe framework with the government are at least as high as the ones prevailing when a safety net is ruled out from the outset. One must bear in mind that the government’s presence is not synonymous of an intervention taking place but only that the government will step in if social welfare - the sum across agents of the return on thei r investments - is to cross a certain threshold in case no action is taken. As it turns out, this possibility alone is enough to make prices to be higher in the government’s framework compared to the alternative, regardless of the informational scenario at hand. Providing the conditions for higher prices, however, does not allow the government’s safety net to sustain just any price: given the measure of social welfare, the higher the price, the higher the likelihood of an intervention: after all, investments are less likely to be profitablewhen theprice paid for them, i.e., their cost, is high. Naturally, intervention - or, as one might call it, insurance - by the government does not co me for free: even though it alleviates losses, were the investors to know an intervention would ensue they would rather had prefered to abstain from participating in the market, putting all their wealth into the 108 risklessasset. Thisimpliesthatanequilibriumcannotbesustainedinthoseinstanceswhere prices are too high and, therefore, a government intervention is imminent. In terms of comparative statics, some nontrivial results come to the fore. First, since losses are limited in the presence of the government’s safety net, one might be induced to think that increases in uncertainty make buying the asset more attractive, with a cor- responding positive effect on prices. Not only that, however, turns to be important, as increases in uncertainty also lead investors to put more weight on the side of an interven- tion: as discussed in the previous paragraph, in this case the bestdecision would beto save the entire wealth for future consumption, instead of choosing to buy the asset. Ultimately, it is the supply level that happens to be pivotal in determining which of these two effects is more important: if the supply is low, market clearing implies few agents participating in the market and, accordingly, social welfare would not be significantly exposed to large swings in the asset’s payoff adventing from changes in uncertainty - the equilibrium price ends up increasing. With a high supply, on the other hand, social welfare is more suscep- tible to change even after marginal increases in uncertainty, and so is the likelihood of an intervention - the equilibrium price ends up dereasing. Second, an increase in the wealth level does not necessarily lead to an increase in the equilibrium price, as one would reasonably expect. The reason is that heterogeneity in the modelcomesnotfromthedistributionofwealthbut,rather,fromthedistributionofsignals. Every agent being endowed with the same wealth, the marginal buyer is by definition the one who, conditional on the signal received, is indifferent between buying or not the asset. In other words, an increase in wealth does not have the effect of increasing the equilibrium price by changing the marginal buyer: that is still defined solely by the distribution of signals. In fact, increasing wealth might even decrease the asset’s equilibrium price, since by not investing agents can secure their entire endowments for future consumption - saving might become a better option. 109 Regarding the functional form of the equilibrium prices, one factor that turns to be important is the unitary downside risk associated with the risky investment. Upon the decision of buying the asset, this measures gives how much of a loss one would face in a worstcase scenario -i.e., withgovernment intervention -r elative to theoption of saving the entire wealth for future consumption. By definition, the downside risk is closely related to thegovernmentsafetynet: thehighertheformer,thelowerthelater. Itistherelationofthe parameterdefiningtheinvestor’sdegreeofuncertaintyand thedownsideriskthatdefinesthe functionalformoftheequilibriumprice: inthelow uncertainty case, thesafety netdoesnot appear in the equilibrium price, regardless of the framework; with high uncertainty, on the otherhand,governmentinsurancebecomesexpliciteveninthosecaseswhereinequilibrium an intervention does not take place. The main motivation of the paper, as it can be grasped from the introduction, is the 2007-2009 financial crisisthathittheworld economy. Oneof thedebatesback intheperiod -whichstillseemstopersistasanopenquestion-referstoh owcanagovernmentsalvageits economy without exposing itself to the problem of moral hazard. The too-big-to-fail policy creates incentives to institutions to behave recklessness: beingaware they cannot besimply let go, they allow themselves to indulge in all sorts of investments. At the same time, since a big chunk of the systemic risk an economy faces is represented by the failure of those very same institutions, they cannot be allowed to go under water indeed: the government has to step in to prevent a collapse with more serious consequences. As it is, the model cannot address questions that deal with the strategic choices of financial institutions entertaining the possibility of a bailout in times of distress. Several papers already do this, among them the few cited in the related literature section. Rather, the aim is to study how the government, by means of the pursued policies, might end up sowing the seeds of a future crisis that is its intention to avoid in the first place. The effect on prices of government’s action seems to be a good place to start: with the possibility 110 of intervention commanding higher prices, if the asset’s payoff cannot keep up with the increased cost of the investment, it is more than natural to expect losses to ensue, which would cause a negative shock in the demand and, consequently, a decline in prices. In extreme cases, these mechanics could generated the boom-bu st patterns that are often associated with (financial) crises. Few remarks about extensions and next steps are: • Thewaythegovernment’ssafetynetinduceshigherpricesin themodelisbyattracting investors thatotherwisewouldprefernottobuy-sincethea ssetisinfixedsupply,the only way the market can accomodate new participants, i.e., for the market to clear, is through a price increase. To see how this process would evolve requires a dynamic model, which is not the case here; • Wealth, e.g. availability of credit, seems to be important in fueling those very same bubblesthat go hand-in-handwith financial crises and, stil l, themodel is silent about that. Introducing heterogeneity in the endowment of investors would allow one to study the effects on prices of changes in the distribution of wealth 19 ; • The social welfare measure adopted includes, in fact, only the welfare of investors. The model abstracts from the welfare of the government itself, without providing any clue of what the optimal level of intervention should be; • Fromtheresultsobtained,followingaresometestablehypothesisthatcouldbeverified empirically 20 : – In comparing assets that otherwise would be deemed with the same character- istics, those with an implicitly government guarantee command a price higher 19 Dealing with two distributions - one for wealth and another f or the signals - might be trick so in a first step the assumption could be that wealthier agents are also better informed. 20 A recent empirical paper related to the themes discussed here is Kelly, Lustig, and Van Nieuwerburgh (2011). 111 than those without; – In cases where it is credible to count on the government’s safety net, increasing the degree of uncertainty leads to a price increase for assets in low supply and a price decrease for those in high supply; – Situations of high prices that make the government extremely susceptible to intervene are followed by price disruptions. 112 Chapter 3 Government Induced Bubbles This chapter builds a model of bubble inflation based on Morris and Shin (1998), with investors decidingwhetherornottobuyanassetthatentails theriskofacollapse inprices, in which case only government intervention could make them avoid a substantial loss. The more investors decide to buy, the more the bubble inflates, and government intervention takes place only when the collapse in prices is sufficiently large. In the benchmark scenario of common knowledge, self-fulfilling beliefs lead to multip licity of equilibria. Using a global games approach, the introduction of a small noise in the signal received by the speculators yields a unique equilibrium, with intervention occuring only if the state of fundamentals happens to be higher than a particular threshold. In a comparative static exercise, it is shown that the government is more likely to step in and bubbles be large the less liquid the asset and the higher the aggregate wealth of investors. 3.1 Introduction Bubble is the term used in the economic literature to define situations where the price at which an asset is transacted does not correspond to the one that would be dictated by its fundamentals, i.e., the factors that determine its payoff - the so-called fundamental price. 113 In case there is one, the size of a bubble could be obtained by comparing the fundamental price to the price that prevails in the market 1 . To assert that there is a bubble in a particular asset or market is not an easy task, though. For the fundamental price depends on the model adopted and, in case there is a discrepancy between the observed price and the fundamental one, it could be argued that the model being used is misspecified or, indeed, not correct. Moreover, one usually has to recur to irrationality assumptions on the part of investors to justify an asset being transacted at a price which is not the fundamental - itself an assumption hard to call for if one whishes to maintain a minimum level of structure in the model. Finally, there is a problem in justifying the existence of bubbles if one believes that there are no arbitrage opportunitiesinthemarket, sincethisimpliesthatinvestorswouldreacttoanymisspricing, pushing the price of the asset back to its fundamental value. Thegoalofthepresentpaperisnottomakeacaseornotfortheexistenceofbubblesbut rathertostudyasituationthatcouldleadtotheiremergence-thepossibilityofgovernment intervention in bust episodes. In particular, the situation we try to depict is the following: investors face the possibility of buying an asset, whose payoff depends only on the state of fundamentals of the economy; the proportion of investors buying the asset makes the observed price to be different from the fundamental one, as if by force of demand pressure; given the existence of a bubble, the state of fundamentals determines if there will be a bust - no if the fundamentals are strong, yes if they are weak; in case of a bust, the government has the opportunity to intervene and absorb part of the drop in prices, trading off the cost ofintevention andthecostof letting thebubbleburst,withthelater beingafunctionof the magnitude of the crash - the difference between the observed and the fundamental pri ce. In case there is a bust and the government decides to intervene, it does so by lowering the interest rate, which increases the fundamental value of the asset and hence reduces 1 For a literature review of bubbles, see Brunnermeier (2007). 114 the size of the crash 2 . We think of a situation where, if the crash is too large, by just letting the bubble burst the government risks the economy dipping into a recession, which ispotentially morecostly thanwhatwouldresultfromtheadoptionof amorelaxmonetary policy - inflation, for example. This cost comparison is what ultimately drives the action of the government in our setup. The investors in turn know that, if the bubble does not burst, the capital gain from the appreciation in prices, regardless of its magnitude, always covers the cost of buying, given by the price paid plus a transaction fee that is meant to represent a liquidity premium 3 . However, in the case of a bust, the same buying decision is justified only when there is government intervention, otherwise staying out of the market would had been the best decision. In the sequence, it is shown that, if the investors were to have perfect information, buying (not buying) would be a dominant strategy for a sufficiently high (low) realization of the state of fundamentals, with multiple equilibria at intermediate levels. This later case of multiplicity of equilibria damages the exercise, as it does not allow one to perform a comparative static analysis - based only on economic reason ing, it is not possible to justify the prevalence of one equilibrium over another when the primitives of the model change. In cases like that, nothing can be said if one is interested in studying the problem from a policy perspective, as we are. As the literature on global games shows, however, we can avoid this multiplicity of equilibria by approximating the original game with one of imperfect information, wherethe stateoffundamentalsisobservedbyeveryone butthegovernment withasmallerror, rather thanbeingcommon knowledge. Inthelimitas theerrorgoes to zero, thegameof imperfect information resembles the original one and, by characterizing the unique strategies’ profile 2 The fundamental value is given by the expected discounted value of the asset’s payoff: decreasing the interest rate decreases the discount factor and hence increases the fundamental price. 3 Liquidityisdefinedastheeasinesswithwhichoneunitoftheassetcanbetransacted,withoutsubstantial price changes. 115 of investors and government that result in equilibrium, the possibility of a comparative static analysis is restored. We view the assumption of investors receiving a signal rather than observing the true realization of the state of fundamentals as a more realistic one, since some degree of noise is always expected to be present in any information dealing with the economy, given the complex nature of the later. The fact that the government keeps its capacity of observing the true state can be viewed as an information differential it has over the investors which, even though debatable, we view not unreasonable of an assumption to have. In this way, framing the original situation as a global game, with investors facing strategic uncertainty, is justified. To yield a unique equilibrium, the global game’s approach bites when it requires the situation being modeled to satisfy some properties, the particularly important one being strategic complementarity on the actions of investors. Strategic complementarity means thatthe(expected)payoffofadoptingonespecificactionisincreasingintheothersadopting that very same action. In our model, it boils down to saying that when all the others are buying (not buying), one particular investor is better off buying (not buying) as well. The strategic complementarity property, not surprisingly, is satisfied in our model, by the following reason: first, investors know that the government, when deciding whether or not to intervene upon a bust episode, trades off the cost of intervention and the cost of the crash itself; second, by buying the asset when all the others also do so, an investor realizes a larger capital gain if the bubble does not burst and, in case it does, she increases the likelihood of government intervention, which is precisely what she wants in this type of situation. This reasoning is reversed in case the buying decision is taken considering that no one else buys. The unique equilibrium that we characterize is of the threshold type, i.e., there is a particular level such that, in a bust episode, if the realization of the state is higher (lower) 116 than that, the government always (never) intervenes. Investors in turn always (never) buy the asset when the signal received is sufficiently high (low). By assumption, the bubble does not burst when the level of fundamentals is sufficiently high and, since that implies thefundamentalsbeingsound,itisnotharmfulforthegovernmenttotolerateabubble-the economy is in a good shape anyway. This is our justificative for the absence of government intervention for those cases where the government acknowledges that there is a bubble but it does not burst 4 . In the comparative static analysis we explore how the threshold equilibrium is altered when the primitives of the model are changed. For instance, we study the consequences of changes inthetransaction cost(liquidity)associated withtheasset; theaggregate wealth of investors; the critical level of fundamentals that determines if the bubbleburstsor not and; the cost of government intervention. By looking to how the equilibrium threshold varies when the primitives of the models change we are able to see under which circumstances government intervention is more likely to occur. According to the results obtained, government intervention is increasing in the trans- action cost of the asset and in the aggregate wealth of investors, while decreasing in the frequency of bubble burts and the cost of intervention. Given that investors have more incentives to buy the asset when it is more likely that the government will intervene, and the higher the proportion of investors buying the asset the more inflated the bubble gets, we can rephrasethe last statement saying that bubbleinflation is increasing in the transac- tion cost of the asset and in the aggregate wealth of the investors, while decreasing in the frequency of bubble bursts and the cost of intervention. Related to the results, first, the bubble getting more inflated the less liquid the asset has a parallel with the latest bubble in the U.S. real state market, whose collapse in the 4 This is a by-product of the fact that the model does not consid er the cost of the economy becoming overheated. 117 beginning of 2007 subsequently triggered a financial crisis. The real state market is no- toriously characterized by assets of low liquidity, which is analogous to a high transaction cost in our model 5 . Second, the positive relationship between the size of the bubble and the aggregate wealth of investors agrees with the usual story according to which an excess of credit helps in fuelling bubbles, by creating an artificial demand that ultimately pushes prices up. Third, increasing the frequency of bubble bursts and the cost of government intervention has more straightforward implications: both make buying the asset less of an attractive choice -capital gains aremoredifficulttorealiz e andthegovernment is lesslikely to be there if needs be - which makes bubbles to be smaller, in t urn diminishing even more the likelihood of government intervention. In case the assumptions of the model are in place, the results mentioned above imply the following testable implications: bubbles should occur more often in less liquid assets; the more credit is available to investors, or the wealthier they are, the more prone should be the economy to bubbles; the more optimistic the investors are about the economy, i.e., theless likely they thinkacrash is tohappen, themorelikely it shouldbetheoccurrenceof bubbles;bubblesshouldbeobservedlessofteninmarketswherethecostforthegovernment to intervene is smaller and; bubbles should be observed less often in economies with weak fundamentals. 3.1.1 Related Literature The paper is mainly related to the literature on bubbles and global games, and it is moti- vated by the latest financial crisis of 2007-09. The idea we ex plore is how the government’s 5 This is not to say that a bubble emerged in the housing market just because of the low liquidity of that particular market. In our model, the less liquid the asset the more costly it is to sell it at the final period, which makes the collapse to be even more pronounced in case of a bust. Since the government intervenes only in large crashes, this implies that there will be more government intervention in less liquid assets, making this type of market more conducive to receive investments and consequently easening the conditions for bubblesto appear. The last bubbleepisode in the U.S.economy being in thehousing market seems to be related to the abundance of credit for investment in that particular market, and not to the fact that bailout guarantees were more explicit there than anywhere else. 118 anticipated policy during a crisis episode might create conditions for a bubble to emerge. In the sequence we mention some papers that touch upon this theme. The financial crisis of 2008-09, as documented in Brunnermei er (2009) and Greenlaw, Hatzius, Kashyap, and Shin (2008), was triggered by the collapse of what was arguably viewed as a bubble in the housing market. Different factors leading to bubbles and boom and bust episodes have been studied by several authors. We cite a few. InGeanakoplos(1997)andGeanakoplos(2003), itisdevelopedtheleveragecycletheory, according to which boom and bust episodes are due to the leverage conditions available to investors. When the economy is doing well, there is a loosening in credit conditions that leadstoahigherleverageofinvestors’portfolios,producingaboom. Whenthefundamentals of the economy deteriorate, the very same investors, for different reasons (e.g., margin requirements), are forced to deleverage their positions, making assets to change handsfrom people who appreciate them more to people who appreciate them less, which depresses prices. Therefore, times whenleverage isallowed toincrease lead tobooms(andpotentially bubbles), with busts happening when investors are required to reduce their exposition to risk due to a worsening in the economy’s fundamentals. Along similar lines, Mendoza (2010) builds an equilibrium business cycle model with a collateral constraint, introducing in the model a Fisherian debt-deflation mechanism: a sufficiently high leverage ratio leads to the fire-sale of asse ts in case the economy is hit by a negative shock, which causes prices to fall and consequently tightens the collateral constraint even more, leading to further fire-sales and so on . It is again the drivers of the leverage ratio of investors that are deemed responsible for episodes of boom and bust. Brunnermeier and Sannikov (2010) develop a macroeconomic model with a financial sector and, rather than just analyzing the behavior of the system around the steady state, provide the full dynamics of the economy, showing how it oscilates between episodes of high and low volatility. They explore the interplay between the economy’s financial and 119 real sector, focusing on the externality the former imposes on the later. It is by neglecting this externality that the financial sector gets too leveraged, leading to an amplification of negative shocks. The connection between market liquidity and funding liquidity is the relevant issue in Brunnermeier and Pedersen (2009). In their model, market liquidity refers to how easily an assetcanbetransacted, withfundingliquiditybeingrelatedtotheavailability offundingfor trading by speculators. The authors show under which circumstances the economy might face liquidity spirals, the process by which a loss suffered by speculators leads to funding problems, increasing margins and subsequently decreasing the market liquidity of assets, depressingpricesandmakingthelossesevenmorepronounced,whichinturnfeedbacksinto furtherdecreases infundingliquidity. Itisinthis sensethattheauthorsclaim thatmargins can be destabilizing. The model offers several new testable implications demanding further empirical investigation. Arguably, one important factor triggering boom and bust episodes is the coordination motive of investors, in the following sense: if there is a perception that a good investment opportunity is available on the market, by coordinating to buy altogether investors might produce a boom; if the sentiment is that an asset currently held does not have positive prospects anymore, investors might want to sell it, which might cause prices to collapse if done jointly. Importantly, the perception of a good investment opportunity depends not only on the research made by the investor herself but also on what she thinks the others think of theasset, i.e., her beliefs about the beliefs of the others, after all it is morerealistic to assume that agents have different information sets. One consequence of investors following strategies that depend on their beliefs is that, in casethereisamechanism devicebywhichactions mightbecoordinated, multipleequilibria canbesupported. ThisisforcefullyexemplifiedintheseminalworkofDiamondandDybvig (1983) on bank runs, where both a “good” and a “bad” equilibri um are supported: the bad 120 one, for example, happens when investors anticipate that others will panick and run to the bank, leading them to choose to run as well, materializing the bank run that originally was only a hypothesis in investors’ minds. Several other economic situations can be framed in a similar setup, as exposed in Cooper (1999). Morris and Shin (2000), however, argue that this indeterminacy of beliefs leading to multiple equilibria is not grounded in economic terms but it is rather a consequence of two modelingassumptionsusually made: common knowledge of fundamentalsand agents being assumed, in equilibrium, to be certain about each other’s behavior. A public signal about the fundamentals, once assumed to be common knowledge, for instance, might constitute the mechanism device that allows agents to perfectly coordinate their beliefs and actions, in a way that, as the authors put it, invites to a multiplicity of equilibria. The global games approach, atermcoined inCarlssonandvan Damme(1993), provides a way around this problem. The basic idea is to add a small noise to the signals to be received by the agents, breaking down the assumption of common knowledge. Without that assumption, agents are not sure about what each other knows, i.e., the information each other has. This uncertainty ends up restricting the set of possible actions to a unique optimal one. Morris and Shin (2003) exposes the theory behind the global games approach and discuss some applications. Several papers use the global games approach to the study of financial crises of some sort. Morris and Shin (2009) compares the portion of credit risk that is due to insolvency, i.e., when the payoff of an asset falls short of the obligation to be paid, to that resulting from illiquidity, i.e., when default in turn is due to a lack of funding that otherwise would allow the borrower to carry on the project, which can occur despite the investment being profitable. This gives rise to situations where an illiquid but otherwise solvent institution defaults (aka Bear Sterns). He and Xiong (2010) develop a similar idea but in a dynamic context, showing that, by spreading the maturity of debt obligations over time, borrowers 121 create a dynamic coordination problem to lenders, potentially leading to preemptive runs. Guimarães (2006) studies currency attacks, in a framework where agents have to decide when to act and, due to market frictions, bear the risk of being caught by a devaluation. In Abreu and Brunnermeier (2003), agents become aware sequentially of the existence of a bubble, and at any point in time there is uncertainty regarding the fraction of speculators who ackowledge it. Speculators then might prefer to ride the bubble and realize one extra period of capital gains, instead of cashing in the profits immediately, bearing the risk of a bust in between. This leads bubbles to persist, regardless of the presence of arbitrageurs. The paper we draw heavily on is Morris and Shin (1998), where a static model of currency attack is developed. In their model, speculators obtain a positive gain in case they attack the currency and the government is forced to abandon the peg, which happens whenever the cost of defending is larger than the cost of letting the currency float. The cost of mantaining the peg is decreasing in the state of fundamentals and increasing in the number of speculators, as economic intuition would dictate. In the benchmark case where the state of fundamentals is common knowledge, a multiplicity of equilibria arises, due to the self-fulfilling nature of beliefs, whereas if the specul ators observe the state with a small noise, this multiplicity reduces to a unique equilibrium, with speculators attacking if and only ifthesignal received is sufficiently small -thatindica tes, after all, thatthegovernment is in a weak position to defend the peg. Among other things, the unique equilibrium obtained in Morris and Shin (1998) is interestingbecauseitallowsonetoperformacomparativestaticanalysisand,inourcontext, to study which factors are more conducive to the emergence of bubbles. This is the main motivation behindusfollowing Morris andShin’sapproach. Inparticular, as intheir model speculators attack the currency based on (i) their perception of the government’s cost of defending the peg and (ii) the fraction of others attacking, in ours agents buy an asset and fuel a bubble in accordance with (i) what they perceive to be the cost for the government 122 to intervene and (ii) the fraction of other speculators who also want to invest. One of the main debates throughout the financial crisis of 2007-09 was regarding the participation of the government in the bailout of instituions that suffered prominent losses, whose eventual collapse would supposedly represent a serious threat to the stability of the entire financial system. One view is that, with the implicit government support of the so- called“too-big-to-fail” institutions, incentivesarecr eatedsothatmorerisksaretaken, since profits are privately gained and losses are public shared 6 . Although not exactly with the same flavor, the assumption of government intervention absorbing crashes is made exactly trying to capture this type of situation, in order to see if the possibility of government participation during crises would result in speculators buying the asset more often, making the economy more prone to bubbles. As it turns out, this is indeed what happens in our model. The structure of the paper is as follows: in the next section we detail the model, first analyzing the benchmark case of perfect information about the state of fundamentals; fol- lowing, we assume that speculators only receive a signal about the state of the economy and, with that assumption in place, the unique equilibrium is derived, with an exercise in comparative static analysis being performed subsequently; finally, the last section digresses on some of the testable implications of the model. 3.2 Model Consider a 1-good, 3-period economy, t = 0,1,2, populated by a unit mass of agents, to be called speculators, represented by the set I = [0,1] 7 . At the initial date, t = 0, each speculator can take one of two actions, buy or not the asset, that’s to say, each i∈ I can either choose X i 0 = 1 (buy) or X i 0 = 0 (not buy). If a speculator decides to buy the asset 6 The bailout money, after all, usually comes from tax payers. 7 We use the words good and asset interchangeably. 123 she pays its initial price, p 0 , plus a fee that represents a transaction cost, t, to be incurred at t = 2 when the asset is sold. If she decides not she incurs no cost. Each speculator can buy at most one unit of the asset. After the speculators make their decisions, in period t= 1, the interim stage, the price of the asset adjusts to p 1 ≡ g(θ,r,α), where θ is a variable that represents the state of fundamentals of the economy, r is the prevailing interest rate and α is the fraction of speculators who buy the asset. We think of g(·) as the observed price of the asset at the interim stage, with the following assumption 8 : Assumption. g θ >0, g r <0 and g α >0. The assumption above states that the observed price is increasing in the fundamentals of the economy, decreasing in the interest rate and increasing in the fraction of speculators who buy the asset. If we think of the asset as being an investment whose performance is better the better are the fundamentals of the economy, it is natural to assume that g(·) is increasing in its first argument. Because the price represents the present value of futurepayoffs, an increase in the interest rate shoulddecrease it. Theprice beingpositively dependent on the fraction of speculators buying the asset represents the effect of demand pressure. The observed price, given by the function g(θ,r,α), must not necessarily be equal to the fundamental price of the asset, which we denote byf(θ,r). Theimportant difference is that, in our interpretation, the fundamental price should not be a function of the fraction of investors buying the asset, since the asset itself represents the present value of a future payoff that is independentof the numberof buyers. With theobserved pricebeingdifferent from the fundamental one, we define a bubble by: b(θ,α)≡g(θ,r,α)−f(θ,r). (3.1) 8 The subscript represents the partial derivative of the function with respect to that argument. 124 We concentrate on positive bubbles, and we also assume that, when the fundamentals improve, the observed price increases more than the fundamental value, which could be seen as a result of an increase in the confidence of the speculators, whose actions have an effect on the observed price but not in the fundamental value. In other words, we have: Assumption. For any θ, r and α, the functions f(·) and g(·) are such that g(θ,r,α) ≥ f(θ,r). Also, b θ >0. Analogously to the assumption made for the function representing the observed price, we impose that: Assumption. f θ ≥0, f r <0. At the interim stage, if the fundamentals of the economy are strong enough (strong economy), that is to say, if θ >L, for a specific L, the bubble does not burst and the price of the asset in period t = 2 equals the price at the interim stage, g(θ,r,α). If the bubble doesburst(weakeconomy), θ≤L,thepriceinthefinalperioddependsonthegovernment’s market intervention decision. We model the government behavior assuming that it faces a costequaltothesizeofthebubbleincaseitburstsandthegovernmentdoesnotinterverne, causing the price to drop from g(θ,r,α) to the fundamental one, p 2 = f(θ,r). Otherwise, by intervening the government pays the cost c to set the interest rate at r ∗ < r, making the price to drop to p 2 = f(θ,r ∗ ) > f(θ,r) instead. Therefore, the investors who buy the asset always benefit when the government decides to intervene in a bust episode, and the government does so whenever b(θ,α)≥c. We can think of the above specification in the following way: if the economy is strong enough, the bubble has no significant cost for the government, after all the fundamentals are sound and the fact that there is a positive bubble represents no significant threat. On the other hand, that is to say, if thefundamentals are relatively weak andthere is a bubble, θ ≤ L, it bursts and the drop in the price level can be sufficiently large, b(θ,α)≥ c, such that, if it realizes, the economy faces the risk of dipping into a serious recession, which 125 would cause a large cost for the government. If that is the case, the government prefers to intervene in the market, lowering the interest rate and therefore absorbing some of the impact of the bust. The parameter c can be viewed as a cost associated with the adoption of a laxer monetary policy, like inflation, after all the government reduces the interest rate when it intervenes in the market. Given the speculators’ decisions and the government’s acti on, the payoff (return) to a speculator at period t=2 is: R i X i 0 ,θ,r ≡X i 0 (p 2 −p 0 −t)=                X i 0 [g(θ,r,α)−p 0 −t] if no bust X i 0 [f(θ,r)−p 0 −t] if bust & w/o govt X i 0 [f(θ,r ∗ )−p 0 −t] if bust & w/ govt (3.2) where no bust representsthecase whenthebubbledoesnotburst,θ >L, and, inabust episode, θ≤L, w/o govt means that the government does not intervene, with no change in the interest rate, while if the government intervenes, w/ govt, it sets the interest rate at r ∗ . With the specification above, we proceed to analyze the equilibrium that would result in the economy under two different scenarios: first when the speculators have perfect infor- mation about the state of fundamentals, θ, and then assuming they receive only a signal x about it, while the government can still observe the true value. As we will see in the sequel, with perfect information, for a certain range of parameters, there is multiplicity of equilibria, whereas in the case of a small level of uncertainty about the fundamentals the resulting equilibrium is unique, allowing us to perform comparative static analysis. 3.2.1 Perfect information about the fundamentals We start assuming that, in the economy specified above, the speculators have perfect in- formation about the parameter θ, representing the fundamentals of the economy. The 126 parameter L is also assumed to be common knowledge. We can think of the following scenario: 1. Nature draws θ according to a uniform distribution, θ∼U[0,1] 2. Government and speculators observe θ 3. Speculators choose to buy or not the asset 4. Government observes therealized proportionof speculatorswhobuytheasset, α, and decides to intervene or not 5. Game ends (investors sell the asset). An equilibrium is a profile of strategies for the speculators and the government such that, in equilibrium, none of them has an incentive to deviate. A strategy for a speculator is a rule that assigns a specific acion for each possible realization of the fundamentals, θ, while for the government is the decision of intervening or not in the market for all possible combinations of the fundamentals and the fraction of speculators who buy the asset, θ and α. We can solve for the equilibrium in this model by proceeding backwards. First we determine the optimal strategy of the government, given the fundamentals and the action of thespeculators, and, basedon that, wedeterminetheoptimal strategy ofthespeculators themselves. Following Morris and Shin (1998) we make some additional assumptions in order to make the problem economically interesting: Assumption. The functions representing the observed price and the fundamental value, g(θ,r,α) and f(θ,r), respectively, are such that the induced bubble b(θ,α) satisfies the following: • b(L,0)<c • b(0,1) <c 127 • b(L,1)>c. An illustration of assumption 3.2.1 is given in the first graph of figure 3.1. The first above, b(L,0) < c, implies that, when the speculators do not buy the asset, that is to say, for α =0, the cost for the government to intervene in the market is always larger than the cost of the bubblebust itself, either if, conditional on a bust, the economy is at its worst or beststate of fundamentals,θ =0orθ =L, respectively 9 . We couldsay that oneway abust episode can severely impair the economy, and hence be very costly for the government, is by affecting the confidence of investors who ultimately suffer the losses caused by the price drop. If the fraction of investors in the market is negligible, or if the market in question does not represent a significant fraction of the economy, with α=0 being a proxy for both situations, then the above says that the government does not intervene. The second assumption made above is that, in the worst state of fundamentals given a bubble burst, θ = 0, the bubble is never large enough to justify government intervention, even if all the speculators are long on the asset, α = 1, which corresponds to b(0,1) < c. This tries to capture the idea that, if the fundamentals are too poor, a bubble never gets too large just as a byproduct of the action of speculators going long on the asset, that is to say, if the economy is too weak the speculators cannot have any hope to inflate the price of the good just by putting demand pressure on it. The last one, b(L,1) > c, represents the fact that, in a bust episode, even when the fundamentals are at their best, θ = L, if there is too much demand pressure, α = 1, the bubble burst becomes too large to be tolerated, after all the economy in this scenario is weak, θ≤L. The government then intervenes in the market. Following Morris and Shin (1998), let us denote by θ the value of θ which solves, for θ≤L, the equation b(θ,1)=c. In other words, θ is the value of θ at which the government 9 Remember that b θ > 0 and, therefore, b(L,0)< c implies that b(0,0)<c. 128 is indifferent between intervening or not in the market when all the speculators are long on the asset, that is, α=1. We assume the following: Assumption. The functions representing the observed price and the fundamental value, g(θ,r,α) and f(θ,r), together with the initial price and transaction cost of the asset, p 0 and t, satisfy: • f(θ,r)<p 0 +t • f(θ,r ∗ )=p 0 +t • g(L,r,α)≥p 0 +t. Hence, if there is no government intervention, the fundamental value in a bust episode is always less than the total cost for the speculator to be long on the asset. On the other hand, the speculator always benefit from government intervention when θ∈ [θ,L], since in that range f(θ,r ∗ )>p 0 +t. The last above means that, in case the bubble does not burst, the observed price is always larger than the total cost of buying the asset. An illustration of assumption 3.2.1 is given in the second graph of figure 3.1. With the above assumptions in place, for the case when θ is common knowledge we have: • θ∈[0,θ]: thecost of intervention is larger than justletting thebubbleburst, sincewe know that b θ > 0 and b(θ,1) = c. No matter what the government’s decision is, the total costforanspeculatortobuytheassetisalways largerthanthemaximumreturn thatshecouldobtain. Therefore, inthisrangethegovernment doesnotintervene and is a dominant strategy for each speculator to not buy the asset, which makes “not buy” (speculators) and “not intervene” (government) to be a n equilibrium; 129 • θ ∈ (θ,L]: the cost of intervention is less than the bubble burst cost, provided that a sufficiently large number of speculators buy the asset. In particular, if none of the speculators buy the asset, the bubble cost is less than the cost of intervention, as mentioned before, and hence the government will not intervene in the market, which in turn justifies the decision of not buy of the speculators in the first place, making “not buy” (speculators) and “not intervene” (government) t o be an equilibrium. On the other hand, since by assumption f(θ,r ∗ ) > p 0 +t for any θ ∈ (θ,L], that is, a speculator will make a positive profit if the government is to intervene when θ lies in thisspecificrange,andthefactthatinthiscasethegovernmentalwaysintervenesince b(θ,1)>c, “buy” (speculators) and “intervene” (government) is also an equilibrium. Analogously tothenomenclatureusedbyMorrisandShin(1998), wenamethis range ripe for intervention; • θ ∈ (L,1]: the fundamentals are sound and the bubble neither bust nor represents a threatfortheeconomy, requiringnogovernmentintervention. Sinceg(θ,r,α)>p 0 +t for any level of the fundamentals in this range, it is a dominant strategy for the speculators tobuytheasset, makingthepair“buy”(specula tors) and“notintervene” (government) to be an equilibrium. Summarizing, if the fundamentals are too poor, θ ∈ [0,θ], or sufficiently good, θ ∈ (L,1], the government does never interverne, “not buy” being a d ominant strategy for the speculators in the first case and “buy” in the second. Figure 3 .1 provides an illustration of the resulting situations according to the state of fundamentals realized, θ. Following Morris and Shin (1998), the interesting range of the fundamentals is the ripe for intervention region. If we suppose that the government makes its intervention decision based only on the trade-off between the cost of inter vention, c, and the cost of no intervention when the bubble burst, b(θ,α), then the fact that the speculators have 130 perfect information regarding the realization of θ gives rise to the standard case of multiple equilibria due to the self-fulfilling nature of the speculat ors’ beliefs, as discussed before. One major problem is that, given this multiplicity of equilibria, no prediction can be made as to whether the investor will end up buying the asset (and hence inflating the bubble) or not. As we will see next, this situation changes when the speculators face a small amount of uncertainty concerning the fundamentals, with each state of fundamentals giving rise to a unique equilibrium. 3.2.2 Imperfect information We now relax the assumption of speculators having perfect information about the state of fundamentals in the economy, θ. In particular, we assume the following structure: 1. Nature draws θ according to a uniform distribution, θ∼U[0,1] 2. Governmentobservesθandeachspeculatorreceivesaprivatesignalx,x∼U[θ−ǫ,θ+ǫ], for some small ǫ 3. Speculators choose to buy or not the asset 4. Government observes therealized proportionof speculatorswhobuytheasset, α, and decides to intervene or not 5. Game ends (investors sell the asset). As in the case with perfect information, we solve for the equilibrium by analyzing first thebehavior ofthegovernment andthendeterminingthereaction functionof theinvestors. We know that the government intervenes in a bust episode only if b(θ,α) ≥ c. Following Morris and Shin (1998), consider the critical proportion of speculators needed to trigger the government to intervene when the state of fundamentals is θ, and let a(θ) denote this critical mass. In other words, for given θ, a(θ) is the value of α that solves b(θ,α)=c. 131 As in Morris and Shin (1998), for a given profile of strategies of the speculators, we denote by π(x) the proportion of speculators who buy the asset when the value of the signal received is x 10 . Also, let s(θ,π) be the proportion of speculators who end up buying the asset when the state of fundamentals is θ. Since x∼U[θ−ǫ,θ+ǫ] we have: s(θ,π)= 1 2ǫ Z θ+ǫ θ−ǫ π(x)dx. (3.3) LetA(π)betheeventwherethegovernmentintervenesifthespeculatorsfollowstrategy π, that is: A(π) ≡ { θ| s(θ,π)≥a(θ)} (3.4) = ( θ| 1 2ǫ Z θ+ǫ θ−ǫ π(x)dx≥a(θ) ) . (3.5) We express the payoff to a speculator of buying the asset at state θ, X i 0 = 1, when aggregate buying activity is π, as: h(θ,π)≡                g(θ,r,α)−p 0 −t if no bust f(θ,r)−p 0 −t if bust & w/o govt f(θ,r ∗ )−p 0 −t if bust & w/ govt. (3.6) However, as mentioned before, the speculators do not observe the true state of funda- mentals, θ, and hence they calculate the expected payoff to buying the asset conditional on 10 Since there is a unit mass of speculator, this can be interpreted as the probability that a speculator chooses to buy the asset when she receives signal x. 132 the signal x, given by: u(x,π) ≡ 1 2ǫ Z x+ǫ x−ǫ h(θ,π)dθ (3.7) = 1 2ǫ " Z [L,x+ǫ] h(θ,π)dθ+ Z [x−ǫ,L]∩A(π) c h(θ,π)dθ+ Z [x−ǫ,L]∩A(π) h(θ,π)dθ # (3.8) wherethesecondlinefollowsfromsplittingthedomainofintegrationinthefirstintegral into three mutually exclusive sets, namely the range for which the bubble does not burst, [L,x+ǫ], and the range in which it does, [x−ǫ,L], which in turn we split in cases when the government intervenes or not, A(π) and A(π) c , respectively. Using the definition of h(θ,π) we can write: u(x,π) = 1 2ǫ " Z [L,x+ǫ] g(θ,r,s(θ,π))dθ+ Z [x−ǫ,L]∩A(π) c f(θ,r)dθ + Z [x−ǫ,L]∩A(π) f(θ,r ∗ )dθ # −p 0 −t (3.9) Since a speculator can guarantee a payoff of zero by refraining from buying the asset, thatis,bychoosingX i 0 =0,therationaldecisionconditionalonsignalxdependsonwhether u(x,π) is positive or negative. Thus, if the government follows its unique optimal strategy, π is an equilibrium of the first period game if π(x)=1 whenever u(x,π)>0 and π(x)=0 whenever u(x,π)≤0. We now enunciate and prove the main result of this section, the analogous version of theorem 1 in Morris and Shin (1998), the unique equilibrium outcome that emerges in the game with imperfect information: Theorem3.1. There is auniqueθ ∗ suchthat, inany equilibriumof the game with imperfect 133 information, conditional on a bubble bust episode the government intervenes if and only if θ ∗ ≤θ≤L. To prove the theorem, we proceed in three steps: first we show that the buyingdecision ofthespeculatorssatisfiesthepropertyof strategic complementarity, thatis, onespeculator is always better off buying the asset whenever the others are doing the same; second, we prove that the function u(x,π) is continuous and monotonic in its first argument; finally, we establish that the equilibrium π is given by a step function, with a unique x ∗ such that the speculators buy the asset whenever the signal they receive is such that x ≥ x ∗ . With these three results is a short step to prove the theorem above. The structure of the proofs follow Morris and Shin (1998). We begin establishing the following: Lemma 3.1. If π(x)≥π ′ (x) for any x, then u(x,π)≥u(x,π ′ ), for any x. Proof. Since π(x) ≥ π ′ (x), we have s(θ,π) ≥ s(θ,π ′ ), for any θ, from the definition of s given by (3.3). Thus, from (3.4) we have: A(π)⊇A π ′ ⇒A(π) c ⊆A π ′ c . (3.10) Inotherwords,theeventinwhichthegovernmentintervesinthemarket, settingr =r ∗ , that is, lowering the interest rate, is strictly larger under π. This and assumptions 3.2 and 3.2 yields: 134 u(x,π) = 1 2ǫ " Z [L,x+ǫ] g(θ,r,s(θ,π))dθ+ Z [x−ǫ,L]∩A(π) c f(θ,r)dθ + Z [x−ǫ,L]∩A(π) f(θ,r ∗ )dθ # −p 0 −t ≥ 1 2ǫ " Z [L,x+ǫ] g(θ,r,s(θ,π ′ ))dθ+ Z [x−ǫ,L]∩A(π ′ ) c f(θ,r)dθ + Z [x−ǫ,L]∩A(π ′ ) f(θ,r ∗ )dθ # −p 0 −t = u(x,π ′ ), which proves the lemma. For the next step, consider the strategy profile where every speculator buys the asset if and only if the message x is more than some fixed number k. Then, aggregate buying activity π will be given by the indicator function I k , defined as: I k =        1 if x≥k 0 if x<k. (3.11) When speculators follow this simple rule of action, the expected payoff to buying the asset satisfies the following property: Lemma 3.2. u(k,I k ) is continuous and strictly increasing in k. Proof. As in Morris and Shin (1998), consider the function s(θ,I k ), the proportion of speculators who buy the asset at θ when the aggregate buying activity is given by the step function I k . Since x is uniformly distributed over [θ−ǫ,θ+ǫ] we have: s(θ,I k )=                0 if k≥θ+ǫ⇔θ≤k−ǫ 1 2 + (θ−k) 2ǫ if k−ǫ≤θ≤k+ǫ 1 if k≤θ−ǫ⇔θ≥k+ǫ 135 where the second line follows from calculating the integral [1/(2ǫ)] R θ+ǫ k dθ. If aggregate short sales are given by I k , there is a unique θ (which depends on k) where the mass of speculators buying equals the mass of speculators necessary to cause the government to intervene in the market, that is, where s(θ,I k ) = a(θ) 11 . Write ψ(k) for theamount thatθ mustbeshortofk for this to betrue. Inother words,ψ(k) is theunique value of ψ solving s(k−ψ,I k )=a(k−ψ). Since the government intervenes in the market if and only if θ lies in the interval [k−ψ(k),L], the payoff function u(k,I k ) is given by: u(k,I k ) = 1 2ǫ ( Z k+ǫ L g(θ,r,s(θ,I k ))dθ+ Z k−ψ(k) k−ǫ f(θ,r)dθ + Z L k−ψ(k) f(θ,r ∗ )dθ ) −p 0 −t. We have: d dk ( Z k+ǫ L g(θ,r,s(θ,I k ))dθ+ Z k−ψ(k) k−ǫ f(θ,r)dθ+ Z L k−ψ(k) f(θ,r ∗ )dθ ) = g   k+ǫ,r,s(k+ǫ,I k ) | {z } =1   + Z k+ǫ L g α (θ,r,s(θ,I k ))(−1/2ǫ)dθ +f(k−ψ(k),r) 1−ψ ′ (k) −f(k−ǫ,r)−f(k−ψ(k),r ∗ ) 1−ψ ′ (k) = g(k+ǫ,r,1)−f(k−ǫ,r)+[f(k−ψ(k),r)−f(k−ψ(k),r ∗ )] 1−ψ ′ (k) − 1 2ǫ Z k+ǫ L g α (θ,r,s(θ,I k ))dθ (3.12) where we have used that fact that, for k−ǫ≤ θ ≤ k+ǫ, s(θ,I k ) = 1/2+(θ−k)/2ǫ, which implies that, for θ = k− ψ(k), we can write 1/2− ψ(k)/2ǫ = a(k−ψ(k)). By 11 Remember that, in a bust episode, for a given θ, a(θ) is the value of α such that b(θ,α) = c; the assumption that b θ > 0 implies that a ′ < 0 and hence there is a unique θ at which s(θ,I k )=a(θ). 136 totally differentiating this last expression we arrive at: ψ ′ (k)= −a ′ 1/2ǫ−a ′ ≥0 (3.13) with the last inequality following from the fact that a θ <0 (see footnote 11). From the above we can also see that for ǫ sufficiently small we have both ψ(k)≈0 12 and ψ ′ (k)≈0, hence 1−ψ ′ (k)≥0. Returningtoexpression(3.12), forsufficientlysmallǫwehavek+ǫ<Lwhich,combined with ψ(k)≈0 and ψ ′ (k)≈0, allows us to claim that: g(k+ǫ,r,1)−f(k,r ∗ ) | {z } >0 +f(k,r)−f(k−ǫ,r) | {z } >0 >0 (3.14) leading us to conclude that u(k,I k ) is monotonically increasing in k. Finally, u(k,I k ) is continuous since is an integral in which the limits of integration are themselves continuous in k. This concludes the proof of the lemma. Before we proceed to the last result necessary to prove theorem 3.1, we discuss the intuition behind lemma 3.2. Recall that when buying activity π is given by the indicator functionI k ,π(x)=1ifx≥k andzerowhenx<k. Thespeculatorsthereforebuytheasset onlywhentheirsignalindicatethatthestateoffundamentalsissufficientlyhigh. Otherwise they risk to be caught in a bust episode where the government does not intervene, which happens when the state of fundamentals is too poor, yielding a negative payoff 13 . Hence, beingmorerestrictiveintheinvestmentdecision,thatistosay,increasingk,guaranteesthat speculatorsbuyonlyatrelatively betterstates, wherebubblesarelesslikely toburstand, if that happens, the crash (bubble) will be large, demanding in turn government intervention and hence increasing the payoff to speculators who buy. 12 The fact that ψ(k) ≥ 0 is implicitly given by 1/2−ψ(k)/2ǫ = a(k−ψ(k)), with a(θ) ≥ 0, implies that for sufficiently small ǫ we must necessarily have ψ(k)≈ 0. 13 See assumptions 3.2.1 and 3.2.1. 137 Put another way, when the fundamentals of the economy are stronger, the payoff to buying the asset is higher for a speculator on the margin from not buying to buying the asset. Analogously to the discussion in Morris and Shin (1998), such a property would be a reasonable feature of any model of bubbles where the bubble cannot be strong (large) enough if the fundamentals are weak and the government intervenes in the market only in cases of large busts. We now establish the last resultnecessary to prove the theorem above: Lemma 3.3. There is a unique x ∗ such that, in any equilibrium of the game with imperfect information of the fundamentals, a speculator with signal x buys the asset if and only if x>x ∗ . Proof. Again following Morris andShin(1998), toprove theabove webegin byestablishing that there is a unique value of k at which u(k,I k )=0. (3.15) From lemma 3.2, we know that u(k,I k ) is continuous and strictly increasing in k. If we can show that it is positive for large values of k and negative for small ones, then we can guarantee that u(k,I k )=0 for some k. When k is sufficiently large (i.e., k≥ L+ǫ), the speculator knows that the bubble will notburst,sincesuchamessageisconsistentonlywitharealization ofθ intheinterval[L,1]. Since the payoff to buying the asset is always positive when the bubble does not burst 14 , we have u(k,I k ) > 0. Similarly, when k is sufficiently small (i.e., k ≤ θ−ǫ), the marginal speculator with message k knows that such a message is consistent with a realization of θ only in the interval [0,θ], region where, as argued before, the cost of intervention for the government is always larger that the one it incurs by just letting the bubble burst, even if all the speculators are on the buy side, that is, α=1, which yields u(k,I k )<0. Therefore, 14 See assumption 3.2.1. 138 there is a unique value of k for which u(k,I k ) = 0. We define the value x ∗ as the unique solution to u(k,I k )=0. Consider now any equilibrium of the game and denote by π(x) the proportion of spec- ulators who buy the asset given message x. Define the numbers x and x as: x ≡ inf{ x| π(x)>0} (3.16) x ≡ sup{ x| π(x)<1} . (3.17) We then have: x≥sup{ x| 0<π(x)<1}≥ inf{ x| 0<π(x)<1}≥ x, (3.18) where the first inequality follows from the fact that{ x| 0<π(x)<1}⊆{ x| π(x)<1} and the last from{ x| 0<π(x)<1}⊆{ x| π(x)>0} . Therefore, x≤x. (3.19) When π(x) < 1, there are some speculators who are not buying the asset. This is consistent with an equilibrium behavior only if the payoff to not buying is at least as high as the payoff to buying given message x. By continuity, this is also true at x. In other words, u(x,π)≤0. (3.20) Now, consider the payoff u(x,I x ). Clearly, I x ≤ π 15 , so that lemma 3.1 (strategic complementarity) and (3.20) above imply that u(x,I x )≤u(x,π)≤ 0. Thus, u x,I x ≤0. 15 Since the indicator function is just a particular case of π and by the definition of x. 139 Since we know from lemma 3.2 that u(k,I k ) is increasing in k and x ∗ is the unique value of k which solves u(k,I k )=0, we have: x ∗ ≥x. (3.21) Following thesamelineofreasoning, whenπ(x)>0, thereareatleastsomespeculators buyingtheasset. Thisisconsistentwithanequilibriumbehavioronlyifthepayofftobuying is at least as high as the payoff to not buying given message x. By continuity, this must also be true at x. In other words, u(x,π)≥0. (3.22) Now, consider the payoff u x,I x . Clearly, I x ≥ π so that lemma 3.1 and (3.22) imply that u x,I x ≥ u(x,π) ≥ 0. Thus, u x,I x ≥ 0. Since we know from lemma 3.2 that u(k,I k ) is increasing in k and x ∗ is the unique value of k which solves u(k,I k ) = 0, we have: x≥x ∗ . (3.23) Thus, from (3.21) and (3.23), we have x≥x ∗ ≥x. However, from (3.19) this implies x=x ∗ =x. (3.24) Thus, the equilibrium π is given by the step function I x ∗, which is what lemma 3.3 states, completing the proof. Following Morris and Shin (1998), it is a short step to the proof of the main result, theorem 3.1. Given that the equilibrium π is given by the step function I x ∗, the aggregate 140 buying activity at the state θ is given by: s(θ,I x ∗)=                0 if θ≤x ∗ −ǫ 1 2 + (θ−x ∗ ) 2ǫ if x ∗ −ǫ≤θ≤x ∗ +ǫ 1 if θ≥x ∗ +ǫ. (3.25) Aggregate buyingactivity s(θ,I x ∗)is increasing inθ, as itis clear from theabove, when its value is strictly between 0 and 1, while a(θ) is decreasing in θ, as argued before 16 . Figure 3.2 below illustrates the derivation of the cutoff point at which the equilibrium buying activity is equal to the level that prompts government intervention. We know that x ∗ > θ−ǫ, otherwise not buying the asset is a strictly better action, contradicting the fact that x ∗ is a switching point. Thus, s(θ,I x ∗) and a(θ) cross precisely once. Define θ ∗ to bethevalue of θ at which these two curves cross. Then, s(θ,I x ∗)≥a(θ) if and only if θ≥θ ∗ , so that the government intervenes in the market if and only if θ≥θ ∗ , which is the claim of the theorem above. 3.3 Comparative statics and policy implications 3.3.1 Changes in the information structure From theprevioussection, wesaw that, whenthereis nonoise, therearemultiple equilibria throughoutthe“ripeforintervention”regionoffundament als. However, withpositivenoise, there is a unique equilibrium with critical value θ ∗ . The value of θ ∗ in the limit as ǫ tends to zero has the following characterization: Theorem 3.2. In the limit as ǫ tends to zero, θ ∗ is given by 16 See footnote 11. 141 φ ′ 0 (−θ ∗ )[f(φ 0 (−θ ∗ ),r)−f(φ 0 (−θ ∗ ),r ∗ )]+f(θ ∗ ,r)+f(θ ∗ ,r ∗ )=2(p 0 +t). (3.26) Proof. Considertheswitchingpointx ∗ ,whichisthesolutiontotheequationu(x ∗ ,I x ∗)=0. We know that u(x ∗ ,I x ∗) = 1 2ǫ ( Z [L,x ∗ +ǫ] g(θ,r,s(θ,I x ∗))dθ+ Z [x ∗ −ǫ,L]∩A(I x ∗) c f(θ,r)dθ + Z [x ∗ −ǫ,L]∩A(I x ∗) f(θ,r ∗ )dθ ) −p 0 −t and from expression (3.25) A(I x ∗) = { θ| s(θ,I x ∗)≥a(θ)} = θ| 1 2 + θ−x ∗ 2ǫ ≥a(θ) . Now, write 1 2 + θ−x ∗ 2ǫ ≥a(θ) ⇔ ǫ+θ−x ∗ ≥2ǫa(θ) ⇔ ǫ−x ∗ ≥−θ+2ǫa(θ) | {z } ≡Φ(θ,ǫ) and notice that, from 142 Φ θ (θ,ǫ)=−1+2ǫ <0 z}|{ a ′ (θ)<0, (3.27) we conclude that Φ(·,ǫ) admits an inverse function, for fixed ǫ. Denote this inverse function by φ ǫ . Now ǫ−x ∗ ≥Φ(θ,ǫ) ⇔ θ≥φ ǫ (ǫ−x ∗ ), and from this we can finally rewrite A(I x ∗) above as A(I x ∗)={ θ| θ≥φ ǫ (ǫ−x ∗ )} . (3.28) Following (3.28), the domains of integration in the expression for u(x ∗ ,I x ∗) can be expressed as [x ∗ −ǫ,L]∩A(I x ∗) c ={ θ| x ∗ −ǫ<θ <φ ǫ (ǫ−x ∗ )} [x ∗ −ǫ,L]∩A(I x ∗)={ θ| φ ǫ (ǫ−x ∗ )<θ <L} . Using these and the fact that for ǫ sufficiently small we have x ∗ +ǫ<L, u(x ∗ ,I x ∗) can be written as u(x ∗ ,I x ∗)= 1 2ǫ ( Z { θ|x ∗ −ǫ<θ<φǫ(ǫ−x ∗ )} f(θ,r)dθ+ Z { θ|φǫ(ǫ−x ∗ )<θ<L} f(θ,r ∗ )dθ ) −p 0 −t. (3.29) 143 Define F (ǫ)≡ Z { θ|x ∗ −ǫ<θ<φǫ(ǫ−x ∗ )} f(θ,r)dθ+ Z { θ|φǫ(ǫ−x ∗ )<θ<L} f(θ,r ∗ )dθ. (3.30) The condition u(x ∗ ,I x ∗)=0 can be expressed as F (ǫ) 2ǫ −p 0 −t=0. (3.31) Because F (ǫ)→0 as ǫ→0 17 , by using L’Hospital’s rule lim ǫ→0 F (ǫ) 2ǫ = F ′ (0) 2 = f(φ 0 (−x ∗ ),r)φ ′ 0 (−x ∗ )+f(x ∗ ,r)+f(x ∗ ,r ∗ )−f(φ 0 (−x ∗ ),r ∗ )φ ′ 0 (−x ∗ ) 2 = φ ′ 0 (−x ∗ )[f(φ 0 (−x ∗ ),r)−f(φ 0 (−x ∗ ),r ∗ )] 2 + f(x ∗ ,r)+f(x ∗ ,r ∗ ) 2 , where φ 0 (·)=φ ǫ (·)| ǫ=0 . Thus, in the limit as ǫ→0, from expression (3.31) we have: φ ′ 0 (−x ∗ )[f(φ 0 (−x ∗ ),r)−f(φ 0 (−x ∗ ),r ∗ )]+f(x ∗ ,r)+f(x ∗ ,r ∗ )=2(p 0 +t). (3.32) Finally,wenotethatx ∗ convergestoθ ∗ whenǫtendstozero,sinceinthelimits(θ,I x ∗)= I x ∗. 17 Notice that the sets{ θ| x ∗ −ǫ <θ <φǫ(ǫ−x ∗ )} and{ θ| φǫ(ǫ−x ∗ ) <θ <L} can be equivalently writ- tenas{ θ| x ∗ −ǫ<θ < 2ǫ(a(θ)−1/2)+x ∗ } and{ θ| 2ǫ(a(θ)−1/2)+x ∗ <θ <x ∗ +ǫ} ,respectivaly,inthe last using the fact that, for small ǫ, x ∗ +ǫ<L. From this it is clear that both collapse to the empty set as ǫ→ 0, hence F (ǫ)→0. 144 3.3.2 Changes in transaction costs Theorem 3.2 provides a characterization of the cutoff point θ ∗ , the one that determines the states at which the government intervenes in the market. In this section we analyze how the cutoff point θ ∗ changes when t, the variable representing the transaction cost of the asset, changes. Totally differentiating the expression in theorem 3.2 yields: −φ ′′ 0 (−θ ∗ ) dθ ∗ dt [f(φ0(−θ ∗ ),r)−f(φ0(−θ ∗ ),r ∗ )] + φ ′ 0 (−θ ∗ ) f θ (φ0(−θ ∗ ),r)φ ′ 0 (−θ ∗ ) − dθ ∗ dt −f θ (φ0(−θ ∗ ),r ∗ )φ ′ 0 (−θ ∗ ) − dθ ∗ dt =2 ⇔ dθ ∗ dt −φ ′′ 0 (−θ ∗ ) [f(φ0(−θ ∗ ),r)−f(φ0(−θ ∗ ),r ∗ )] + dθ ∗ dt −φ ′ 0 (−θ ∗ ) 2 [f θ (φ0(−θ ∗ ),r ∗ )−f θ (φ0(−θ ∗ ),r)] =2 ⇔ dθ ∗ dt = 2 [−φ ′′ 0 (−θ ∗ )][f(φ0(−θ ∗ ),r)−f (φ0(−θ ∗ ),r ∗ )]+ −φ ′ 0 (θ ∗ ) 2 [f θ (φ0(−θ ∗ ),r ∗ )−f θ (φ0(−θ ∗ ),r)] Now, remember that, by definition, φ ǫ is the inverse function of Φ(·,ǫ), for fixed ǫ, with Φ(θ,ǫ) =−θ+2ǫa(θ). Therefore, Φ θ < 0, which implies that φ ′ 0 < 0. The sign of dθ ∗ /dt is also dependent on the sign of φ ′′ 0 and how f 1 changes for different levels of the interest rate. We impose the following assumption: Assumption. The bubble b(θ,α) is a convex function of θ. Also, f θ f r =0. In the first part of assumption 3.3.2 we impose that the rate at which the bubble is inflated increases as the fundamentals improve 18 . Using the definition of a(θ), this implies thata ′′ <0, whichinturnimpliesthatφ ′′ 0 <0. Thelaststatementintheassumptionmeans thattherateatwhichthefundamentalvaluechangesasthestateoffundamentalsimproveis independent of the prevailing interest rate, yielding [f θ (φ 0 (−θ ∗ ),r ∗ )−f θ (φ 0 (−θ ∗ ),r)] = 18 We can interpret this assumption as an effect of overconfidence: as the fundamentals improve the speculators get more and more optimistic, leading to a buy decision that ultimately makes the bubble to inflate faster the higher the state of fundamentals. 145 0. With this we then have: dθ ∗ dt = 2 −φ ′′ 0 (−θ ∗ ) | {z } >0 [f(φ 0 (−θ ∗ ),r)−f(φ 0 (−θ ∗ ),r ∗ )] | {z } <0 <0. (3.33) Therefore, if assumption 3.3.2 is in place, we conclude that increasing transaction costs decreases θ ∗ , hence increasing the range of government intervention in a bust episode. If we interpret t as a measure of the liquidity of the asset, the higher t the less liquid it is, in equilibrium the government intervenes more in the market the less liquid the asset. This implies that, for sufficiently large x, the signal received by a speculator regarding the state of the fundamentals, investment is more likely to be made on illiquid than liquid assets, whichinturnimpliesthatthegovernment’spolicyfuelsbubblesparticularlyinlessliquidity markets. In a way, the last bubble episode occurring in the housing market, characterized by illiquid assets, offers empirical support for this result. 3.3.3 Changes in aggregate wealth FollowingagainMorrisandShin(1998), weconsidernowhowtheequilibriumcutoffpointθ ∗ changeswhenaggregatewealthvaries. AsMorrisandShin(1998)mention,theinternational flow of so-called “hot money”, or also a more lax monetary poli cy that increases funding liquidity, would constitute factors in determining the level of aggregate wealth, as well as changes in the number of speculators themselves. Changing aggregate wealth would cause an impact on the function a(θ), the critical proportion of speculators that would require government intervention in the case of a bust episode. When the aggregate wealth of the speculators increases, the critical proportion of speculators falls, since the government’s decision to intervene is based on the absolute level of buying activity, and not the number of speculators buying the asset. As we can see in figure 3.2, a downward shift in the function a(θ) decreases the equi- 146 librium θ ∗ and hence has the effect of enlarging the set of states at which the government intervenes in the market in a bust episode, that is, the event A(π) = { θ| s(θ,r)≥a(θ)} gets larger when the function a(·) gets smaller. In this way, since in the case of a bubble burst the payoff to speculators who buy the asset is larger exactly when there is government intervention, the enlargement of the event A(π) has the effect of increasing the (expected) payoff to buying the asset, at any value of thesignalreceived,x. Therefore,withanincreaseinaggregatewealthaugmentingtherange of fundamentals for which the government intervenes, the speculators have more incentives to buy the asset, increasing the incidence of bubbles 19 . 3.3.4 Changes in the threshold level of bubble burst, L A change in L, if it is an increase, makes the bubble more likely to burst, that is, it turns buyingtheasset into less of an attractive investment from the viewpoint of the speculators, since they benefit more when they buy and the bubble does not burst. The speculators buyingless causes bubblesto inflate less, demandingthenalso less government intervention inbustepisodes. Inthisway, anincreaseinLshouldmakeθ ∗ larger, whichiswhathappens if we follow figure 3.2. Analogously, a decrease in L implies that bubbles are less likely to burst, attracting more investors and hence becoming more inflated which requires, at a relatively lower θ, government intervention. Hence, decreasing L should decrease θ ∗ , which can also be seen in figure 3.2. Changes in L can indirectly be interpreted as changes in “market sentime nt”, in the following way: if speculators believe that the true threshold level that separates bubble bursts from other events is given by L ′ rather than L, with L ′ < L, they will be more likely to buy the asset, since from figure 3.2 we can see that if L was to be decreased this would lead to a non-increasing change in x ∗ and, for sufficiently smallǫ, to a non-increasing 19 This pointis made by some authorswho pointto global imbalances as one of themain factors triggering the start of bubbles. 147 change in θ ∗ as well. That means that, at worst, the investors do not change the likelihood of buyingtheasset whenthey think thatthetruethresholdlevel isL ′ , withahigher chance that they will be actually buying the asset more often, that is, adopt the buying decision for a larger range of signals, x. The reason for calling this “market sentiment” is that, if the belief of investors is that the threshold level is L ′ < L, they think that a bubble is less likely to burst than it actually is, since that happens whenever θ < L. The investors could then be said of being overconfident, and here we link this with the concept of market sentiment 20 . 3.3.5 Changes in the threshold level of government intervention, c Changes in c, that is, changing the cost of government intervention, is equivalent to shifts in a(θ), the critical proportion of speculators such that, when they buy, government action is required, in a bust episode. If c increases, that is, if it is more costly for the government to intervene, the bubble will have to be larger to justify such action, hence a(θ) should shift to the right, increasing θ ∗ and hence making the government to intervene less. With lower government intervention and for fixed L, it is less attractive for speculators to buy the asset, which then makes bubblesto be less inflated. An analogous argument imply that a decrease in c leads to a decrease in θ ∗ , which is intuitive since a decrease in the cost of intervention should make a bubble burst relatively more costly, making a better option for the government to intervene. 20 Shiller (2005) discusses the effects of market sentiment on the inflation of bubbles. There, it is argued that, usually, bubbles start to be inflated at periods the population associate with the beggining of so-called new eras, often depicted in a rosy way, indicating that prosperous times came for good, to the benefit of everyone. 148 3.4 Concluding Remarks In this paper we build a model of the onset of bubbles, where government intervention in bust episodes, lowering the interest rate and hence reducing the size of crashes and losses to speculators, plays a key role. The intervention of the government in crises creates more incentives for speculators to buy the asset, resulting in a demand pressure type of effect that inflates bubbles. Using the global games approach, as in Morris and Shin (1998), we are able to derive a unique equilibrium where the government intervenes if and only if the state of fundamentals in the economy is larger than a particular level, which induces the speculators to also adopt a strategy of the thereshold type relative to the signals they receive. Following an exercise in comparative static analysis, we obtain that government inter- vention is higher the less liquid the asset and the higher the aggregate wealth of investors. Given that investors are always better off when there is government intervention and that government intervention depends on the size of the bubble when there is a crash, with the size of the bubble increasing in the fraction of speculators who opt to be long in the asset, weconcludethatbubblesaremorepronetohappeninilliquidassetsandwhencreditafloat. These two results are in a way related to the last bubblebustin the american housing mar- ket, a market characterized by assets of low liquidit and where credit was abundant due to bothglobalimbalancesandtheapetiteofinvestorsforhugegains,symbolizedinthetrading activities of complex derivative instruments. It would be interesting not only to relate our findings to the last financial crisis but rather perform a rigorous empirical analysis to check if such claims are indeed supported by the data. Themodel we present is static butdynamicfactors would need to be considered if more robust results were to be obtained. For instance, if we were to model the government as an institution targeting to control the volatility of the business cycle, with a higher volatility 149 implying a higher cost for the government, in a dynamic model government intervention wouldhappenbothwhenpriceincreasesordecreasestoomuch,withthegovernmentraising and lowering the interest rate in such episodes, respectively. In our model, the government does not “control” the volatility of the business cycle, sin ce the crash that matters is the intra period crash rather than the inter period crash, that is to say, the cost of intervention for the government is a function of the difference between the price at the interim stage and the one that would result after the crash, not the difference between the intial price at t = 0 and the final price at t = 2, which is what actually determines the volatility of the market. 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(A.3) Taking the derivative of H yields 160 H ′ (γ)=− (ρ−ρ ∗ ){ [ρ ∗ +γ(1−ρ ∗ )]+(1−γ)(1−ρ ∗ )[ρ+γ(1−ρ)]} { [ρ+γ(1−ρ)][ρ ∗ +γ(1−ρ ∗ )]} 2 <0, (A.4) the inequality following from ρ>ρ ∗ . For γ≥0, it follows therefore that H(0)≥H(γ), i.e, ρ−ρ ∗ ρρ ∗ ≥ [ρ+γ(1−ρ)]−[ρ ∗ +γ(1−ρ ∗ )] [ρ+γ(1−ρ)][ρ ∗ +γ(1−ρ ∗ )] . (A.5) For γ ∗ ≥γ one has that [ρ+γ(1−ρ)]−[ρ ∗ +γ(1−ρ ∗ )] [ρ+γ(1−ρ)]−[ρ ∗ +γ ∗ (1−ρ ∗ )] ≥1 (A.6) and [ρ+γ(1−ρ)][ρ ∗ +γ(1−ρ ∗ )] [ρ+γ(1−ρ)][ρ ∗ +γ ∗ (1−ρ ∗ )] ≤1, (A.7) therefore [ρ+γ(1−ρ)]−[ρ ∗ +γ(1−ρ ∗ )] [ρ+γ(1−ρ)][ρ ∗ +γ(1−ρ ∗ )] ≥ [ρ+γ(1−ρ)]−[ρ ∗ +γ ∗ (1−ρ ∗ )] [ρ+γ(1−ρ)][ρ ∗ +γ ∗ (1−ρ ∗ )] . (A.8) Combining (A.5) with (A.8) implies that ρ−ρ ∗ ρρ ∗ ≥ [ρ+γ(1−ρ)]−[ρ ∗ +γ ∗ (1−ρ ∗ )] [ρ+γ(1−ρ)][ρ ∗ +γ ∗ (1−ρ ∗ )] , (A.9) which is the desired result. Now one proceeds proving Proposition 1.1: for an even number of banks, N, consider the 161 original network formed without government intervention, γ =γ ∗ =0, and take arbitrarily two of them, say banks i and j. Without loss of generality, assume that r ∗ i −(r i +r j ) ω i −e i > r ∗ j −(r j +r i ) ω j −e j . (A.10) One is interested in what would be the network if, instead, government intervention, γ ∗ ≥γ >0, was in place. From Lemma A.1 it follows that: ρ−ρ ∗ ρρ ∗ > [ρ+γ(1−ρ)]−[ρ ∗ +γ ∗ (1−ρ ∗ )] [ρ+γ(1−ρ)][ρ ∗ +γ ∗ (1−ρ ∗ )] . (A.11) If in the original network banksi and j do not have a link, (1.33) and (A.10) imply that ρ−ρ ∗ ρρ ∗ > r ∗ i −(r i +r j ) ω i −e i . (A.12) Following intervention, however, either ρ−ρ ∗ ρρ ∗ > [ρ+γ(1−ρ)]−[ρ ∗ +γ ∗ (1−ρ ∗ )] [ρ+γ(1−ρ)][ρ ∗ +γ ∗ (1−ρ ∗ )] > r ∗ i −(r i +r j ) ω i −e i , (A.13) which from (1.33) implies that banks still do not want to transact, or ρ−ρ ∗ ρρ ∗ > r ∗ i −(r i +r j ) (ω i −e i ) ≥ [ρ+γ(1−ρ)]−[ρ ∗ +γ ∗ (1−ρ ∗ )] [ρ+γ(1−ρ)][ρ ∗ +γ ∗ (1−ρ ∗ )] > r ∗ j −(r j +r i ) ω j −e j , (A.14) which from (1.28) implies that bank i borrows from bank j and a link is established, or ρ−ρ ∗ ρρ ∗ > r ∗ i −(r i +r j ) ω i −e i > r ∗ j −(r j +r i ) ω j −e j ≥ [ρ+γ(1−ρ)]−[ρ ∗ +γ ∗ (1−ρ ∗ )] [ρ+γ(1−ρ)][ρ ∗ +γ ∗ (1−ρ ∗ )] , (A.15) which from (1.30) implies that both banks want to borrow and that a new link will be 162 created, independently of the identities of the borrower and the lender. Therefore, if in the original network banks i and j do not share a link, with government intervention they will either continue not transacting or instead will create a link. Ifintheoriginalnetworkbanksiandj dohavealoanagreement, then(1.33)and(A.10) imply that r ∗ i −(r i +r j ) ω i −e i ≥ ρ−ρ ∗ ρρ ∗ . (A.16) Following intervention, either r ∗ i −(r i +r j ) ω i −e i ≥ ρ−ρ ∗ ρρ ∗ > [ρ+γ(1−ρ)]−[ρ ∗ +γ ∗ (1−ρ ∗ )] [ρ+γ(1−ρ)][ρ ∗ +γ ∗ (1−ρ ∗ )] > r ∗ j −(r j +r i ) ω j −e j , (A.17) which from (1.28) results in bank i borrowing from bank j and a link is kept in place, or r ∗ i −(r i +r j ) ω i −e i ≥ ρ−ρ ∗ ρρ ∗ > r ∗ j −(r j +r i ) ω j −e j ≥ [ρ+γ(1−ρ)]−[ρ ∗ +γ ∗ (1−ρ ∗ )] [ρ+γ(1−ρ)][ρ ∗ +γ ∗ (1−ρ ∗ )] , (A.18) which from (1.30) means that both banks want to borrow and, no matter what the identities of borrower and lender, they engage in a transaction and a link remains. Therefore, if in the original network banks i and j do not transact, with intervention they might create a link and, if they do transact, they keep transacting and a link keep existing across the banks. By the arbitrariness of banks i and j the result follows. 163 A.2 Proof of Proposition 1.2 Consider the original network with an even number N of banks, being formed without government intervention. Take arbitrarily two banks, say i and j, and assume without loss of generality that r ∗ i −(r i +r j ) ω i −e i > r ∗ j −(r j +r i ) ω j −e j . (A.19) For thesake of theargument, assumealso that iborrowsfromj in theoriginal network, with γ ∗ =γ =0. That means, from (1.28) and (1.30), that either r ∗ i −(r i +r j ) ω i −e i ≥ ρ−ρ ∗ ρρ ∗ > r ∗ j −(r j +r i ) ω j −e j (A.20) holds or simultaneously that r ∗ i −(r i +r j ) ω i −e i > r ∗ j −(r j +r i ) ω j −e j ≥ ρ−ρ ∗ ρρ ∗ (A.21) and [r ∗ i −(1−ω i )−e i −r j ]− h r ∗ j −(1−ω j )−e j −r i i > (ω i −e i )−(ω j −e j ) ρ ∗ . (A.22) Consider first (A.20). From Lemma A.1, one knows that, with government, γ ∗ ≥γ >0, ρ−ρ ∗ ρρ ∗ > [ρ+γ(1−ρ)]−[ρ ∗ +γ ∗ (1−ρ ∗ )] [ρ+γ(1−ρ)][ρ ∗ +γ ∗ (1−ρ ∗ )] . (A.23) Therefore, upon intervention it follows that either 164 r ∗ i −(r i +r j ) ω i −e i ≥ ρ−ρ ∗ ρρ ∗ > [ρ+γ(1−ρ)]−[ρ ∗ +γ ∗ (1−ρ ∗ )] [ρ+γ(1−ρ)][ρ ∗ +γ ∗ (1−ρ ∗ )] > r ∗ j −(r j +r i ) ω j −e j (A.24) or r ∗ i −(r i +r j ) ω i −e i ≥ ρ−ρ ∗ ρρ ∗ > r ∗ j −(r j +r i ) ω j −e j ≥ [ρ+γ(1−ρ)]−[ρ ∗ +γ ∗ (1−ρ ∗ )] [ρ+γ(1−ρ)][ρ ∗ +γ ∗ (1−ρ ∗ )] (A.25) holds. If (A.24) is the case, from (1.28) it follows that bank i continues borrowing from j. If (A.25), bank i will keep borrowing from bank j and not the converse as long as (1.31) is satisfied, i.e., [r ∗ i −(1−ω i )−e i −r j ]− h r ∗ j −(1−ω j )−e j −r i i > (ω i −e i )−(ω j −e j ) ρ ∗ +γ ∗ (1−ρ ∗ ) . (A.26) However, one can show that, if h (r ∗ i −r j )− r ∗ j −r i i ρ ∗ +γ ∗ (1−ρ ∗ ) 1−[ρ ∗ +γ ∗ (1−ρ ∗ )] <(ω i −e i )−(ω j −e j )< r ∗ i −r ∗ j ρρ ∗ ρ−ρ ∗ (A.27) then (A.19) is satisfied whereas (A.26) is not, i.e., without intervention bank i borrows from bank j, and with government the opposite is true - bank i becomes the lender and bank j the borrower. In the second case, i.e., if both (A.21) and (A.22) hold simultaneously, the fact that ρ ∗ <ρ ∗ +γ ∗ (1−ρ ∗ ) implies that 165 (ω i −e i )−(ω j −e j ) ρ ∗ > (ω i −e i )−(ω j −e j ) ρ ∗ +γ ∗ (1−ρ ∗ ) (A.28) and,therefore, (A.22) beingsatisfiedautomatically impliesthat(1.31)alsois, andhence that with intervention bank i is kept as the borrower and bank j the lender. A.3 Proof of Corollary 1.1 As in the proof of Lemma A.1, from (A.8) one has that [ρ+γ(1−ρ)]−[ρ ∗ +γ(1−ρ ∗ )] [ρ+γ(1−ρ)][ρ ∗ +γ(1−ρ ∗ )] > [ρ+γ(1−ρ)]−[ρ ∗ +γ ∗ (1−ρ ∗ )] [ρ+γ(1−ρ)][ρ ∗ +γ ∗ (1−ρ ∗ )] . (A.29) For γ ∗ > γ > 0, the right-hand side of the above inequality is the criteri a for banks to choose between investing in large or small projects. Running through all possible scenarios after banks’ pairwise meetings - expressions (1.28), (1.29 ), (1.30) and (1.33) - one immedi- ately sees that the incentives for both banks to invest in a large project - and hence borrow money - are large with the TBTF policy as opposed to the case wh ere γ ∗ =γ >0. A.4 Proof of Corollary 1.2 Denote by π ∗ ij the profits of bank i with a large project when it borrows from bank j and by π i with a small one, and analogously for bank j. By adding a superscript G one has the same variables but for the case with government intervention. From Proposition 1.1, the number of links across banks do not decrease when a network is formed with intervention, compared to the one obtained without the government. From Proposition 1.2, however, if (A.25) holds, 166 r ∗ i −(r i +r j ) ω i −e i ≥ ρ−ρ ∗ ρρ ∗ > r ∗ j −(r j +r i ) ω j −e j ≥ [ρ+γ(1−ρ)]−[ρ ∗ +γ ∗ (1−ρ ∗ )] [ρ+γ(1−ρ)][ρ ∗ +γ ∗ (1−ρ ∗ )] , (A.30) i.e., without thegovernment bankiborrows from bankj and the last is actually better- off either lending or investing in a small project - π ∗ ij > π i and π j > π ∗ ji - but with inter- vention both banks prefer an investment in a large project - π ∗ ij G >π G i and π ∗ ji G >π G j - it might well be that π ∗ ji G > π ∗ ij G , [r ∗ i −(1−ω i )−e i −r j ]− h r ∗ j −(1−ω j )−e j −r i i < (ω i −e i )−(ω j −e j ) ρ ∗ +γ ∗ (1−ρ ∗ ) , (A.31) which from (1.32) implies that, with government, the role of banks is switched - bank i becomes the lender and bank j the borrower. Given that ρ ∗ +γ ∗ (1−ρ ∗ ) > ρ ∗ , (A.31) implies that [r ∗ i −(1−ω i )−e i −r j ]− h r ∗ j −(1−ω j )−e j −r i i < (ω i −e i )−(ω j −e j ) ρ ∗ , (A.32) which in turn means that, without the government, the profit of bank j with a large project was also bigger than that realized by bank i, or that π ∗ ji > π ∗ ij . Therefore, without government one has that π j >π ∗ ji >π ∗ ij >π i , (A.33) whereas with intervention 167 π ∗ ji G >max n π G j ,π ∗ ij G o and π ∗ ij G >π G i . (A.34) Therefore, in the particular cases of banks i and j, networth increases with government intervention only if π ∗ ji G +π G i >π ∗ ij +π j . (A.35) From π ∗ ji G > π ∗ ij G and π ∗ ij G > π ∗ ij one has that π ∗ ji G > π ∗ ij . If π G > π, then π G i > π G > π > π j leads to π G i > π j and, therefore, (A.35) is satisfied, i.e., intervention leads to a networth improvement - but not necessarily otherwise. Fro m the arbitrariness of banks i and j, thus, the result follows. A.5 Proof of Lemma 1.1 The matrix B is given by: B := I− e X =              1 − r 1(1−α 1 r )χ 12 P k∈N χ k2 r k ··· − r 1(1−α 1 r )χ 1N P k∈N χ kN r k − r 2(1−α 2 r )χ 21 P k∈N χ k1 r k 1 ··· − r 2(1−α 2 r )χ 2N P k∈N χ kN r k . . . . . . . . . . . . − r N(1−α N r )χ N1 P k∈N χ k1 r k − r N(1−α N r )χ N2 P k∈N χ k2 r k ··· 1              . One needs to show that there are positive numbers d 1 ,d 2 ...,d n such that d j |b jj | > P i6=j d i |b ij |, for j =1,...,n, i.e.: 168 d 1 > X i6=1 d i |b i1 | = X i6=1 d i r i 1−α i r χ i1 P k∈N χ k1 r k , d 2 > X i6=2 d i |b i2 | = X i6=2 d i r i 1−α i r χ i2 P k∈N χ k2 r k , . . . d N > X i6=N d i |b iN | = X i6=N d i r i 1−α i r χ iN P k∈N χ kN r k . Suppose that d 1 =d 2 =...=d N =d. The above then becomes: 1 > X i6=1 1−α i r r i χ i1 P k∈N χ k1 r k , 1 > X i6=2 1−α i r r i χ i2 P k∈N χ k2 r k , . . . 1 > X i6=N 1−α i r r i χ iN P k∈N χ kN r k . For any i, one knows that 0 < α i r < 1 or, equivalently, 0 < 1−α i r < 1, which implies that: 1= P i6=1 r i χ i1 P k∈N χ k1 r k > X i6=1 1−α i r r i χ i1 P k∈N χ k1 r k , 1= P i6=2 r i χ i2 P k∈N χ k2 r k > X i6=2 1−α i r r i χ i2 P k∈N χ k2 r k , . . . 1= P i6=N r i χ iN P k∈N χ kN r k > X i6=N 1−α i r r i χ iN P k∈N χ kN r k , 169 as one wanted to show. 170 Appendix B Appendix to Chapter 2 B.1 ComparisonofEquilibriumPricesacrossDifferentFrame- works with Government Intervention For the low uncertainty case, the equilibrium price in the common prior scenario is given by p CP = A−C − t+1− p 2(A−C) - if the wealth constraint is not binding. To be consistent with the possibility of government intervention, the downside risk must be such that 1 0≤A−C ≤1/2. With this assumption in place, the following holds Proposition B.1. For a sufficiently high realization of the state of fundamentals, 1>θ >1+ τ +A−C− q 2(A−C) , (B.1) the ordering of equilibrium prices is p PI >p IP >p CP ; (B.2) for an intermediate realization, 1 Theoriginalassumptiontobemade,followingthediscussionafterexpression(2.25),is0≤ (A−C)/K≤ 1/2. However, recall we are considering the supply level, K, to be unitary. 171 1+ A−C− q 2(A−C) <θ <1+ τ +A−C− q 2(A−C) , (B.3) the ordering is p PI >p CP >p IP ; (B.4) finally, for 0<θ <1+ A−C− q 2(A−C) , (B.5) it holds that p CP >p PI >p IP , (B.6) all the above being true provided the level of uncertainty is low, τ < A−C, and the downside risk being such that 0 ≤ A− C ≤ 1/2, with the equilibrium prices across the different scenarios given by (i) Imperfect information: p IP =θ−τ−t; (ii) Perfect information: p PI =θ−t; and (iii) Common prior: p CP =A−C−t+1− p 2(A−C). Proof. We start with p PI >p IP >p CP . We have p IP >p CP ⇔ θ >1+ τ +A−C− q 2(A−C) . (B.7) 172 Also, τ +A−C <2(A−C)< q 2(A−C), (B.8) where the first inequality follows from the low uncertainty assumption, τ <A−C, and the second from 0 ≤ A− C ≤ 1/2. Therefore, τ + A− C − p 2(A−C) < 0, which in turn implies that 1+ h τ +A−C− p 2(A−C) i < 1. As θ ∈ [0,1], for θ sufficiently high, i.e., θ > 1+ h τ +A−C− p 2(A−C) i , condition (B.7) is satisfied, which is equivalent to p IP > p CP . Finally, since τ > 0, it holds trivially that θ−t > θ−τ −t, i.e., p PI > p IP , which completes the proof of the ordering given in (B.2). For p PI > p CP > p IP , first notice that, from (B.7), θ < 1+ h τ +A−C− p 2(A−C) i implies p IP <p CP . Since from τ >0 it is trivially true that p PI >p IP , all that is left is to compare p PI to p CP . We have p PI >p CP ⇔ θ >A−C +1− q 2(A−C), which implies that, for 1+ A−C− q 2(A−C) <θ <1+ τ +A−C− q 2(A−C) , (B.9) it follows that p PI >p CP >p IP , as in (B.4), whereas for 0<θ <1+ A−C− q 2(A−C) , (B.10) the ordering of prices is p CP >p PI >p IP , as in (B.6), completing the proof. 173 Proposition B.2. In the high uncertainty scenario, τ > A−C, and for a downside-risk such that 0≤A−C ≤1/2, the following holds: (i) If the wealth constraint is binding, the ordering of prices is p CP >p PI >p IP ; (B.11) (ii) If the wealth constraint is not binding, (a) For τ >1/2, a sufficiently high realization of the state of fundamentals, 1>θ >1+A−C− q 2(A−C), (B.12) implies the ordering of equilibrium prices to be p PI >p CP >p IP , (B.13) whereas for a sufficiently low realization, 0<θ <1+A−C− q 2(A−C), (B.14) the ordering is reversed to p CP >p PI >p IP ; (B.15) (b) For τ <1/2, a sufficiently high realization of the state of fundamentals, 1>θ >1+2 q τ(A−C)− q 2(A−C), (B.16) 174 implies the ordering of equilibrium prices to be p PI >p IP >p CP , (B.17) whereas for an intermediate realization, 1+A−C− q 2(A−C)<θ <1+2 q τ(A−C)− q 2(A−C), (B.18) the ordering is p PI >p CP >p IP , (B.19) and finally, in a sufficiently low realization of the state of fundamentals, 0<θ <1+A−C− q 2(A−C), (B.20) the ordering that holds is p CP >p PI >p IP , (B.21) with the equilibrium prices across the different scenarios given by (i) Imperfect information: p IP =θ+A−C−t−2 p τ(A−C); (ii) Perfect information: p PI =θ−t; and (iii) Common prior 2 : p CP =A or p CP =A−C−t+1− p 2(A−C). Proof. Since feasibility requires the wealth to be greater than the price - in case an invest- ment is to be made - it holds trivially that p CP > max n p PI ,p IP o , whenever p CP = A. 2 In the common prior case, p CP =A if the wealth constraint is binding. 175 Comparing p PI to p IP we have p PI >p IP ⇔ τ > 1 4 (A−C). (B.22) In the high uncertainty case, τ > A−C, therefore τ > (A−C)/4, implying condition (B.22) to be satisfied, which is in turn equivalent to p PI > p IP . Therefore, the ordering of equilibrium prices -in case the wealth constraint is bindin g - is p CP >p PI >p IP . If the wealth constraint is not binding, the price in the common prior case is p CP = A−C−t+1− p 2(A−C). Since we know from the above that p PI >p IP , all is left is to compare p CP to p IP and p PI , if need be. First, p IP >p CP ⇔ θ >1+ 2 q τ(A−C)− q 2(A−C) . (B.23) Ifτ >1/2,then2 p τ (A−C)− p 2(A−C)>0,hence1+ h 2 p τ (A−C)− p 2(A−C) i > 1. As θ∈[0,1], it follows that θ <1+ h 2 p τ(A−C)− p 2(A−C) i which, from (B.23), is equivalent to p CP > p IP . Since it was previously established that p PI > p IP , we can write p IP <min n p PI ,p CP o . Comparing p PI to p CP , the equivalent condition is p PI >p CP ⇔ θ >1+A−C− q 2(A−C). (B.24) 176 Theassumption0≤A−C ≤1/2impliesthattheright-handsideof(B.24) isdecreasing in A−C, with 1+A−C− p 2(A−C)∈ [0,1] as a result. Since θ∈ [0,1], (B.24) can be split in two cases: for a sufficiently high θ, 1>θ >1+A−C− q 2(A−C), (B.25) it follows that (B.24) is satisfied, which is equivalent to p PI > p CP , whereas for a sufficiently low θ, 0<θ <1+A−C− q 2(A−C), (B.26) the ordering is reversed to p CP >p PI . In summary, combining the above with the previous result that p IP <min n p PI ,p CP o , thefinalorderingofprices, forasufficiently highlevel offundamentals, isp PI >p CP >p IP , whereas for a sufficiently low that ordering is p CP > p PI > p IP , with both prevailing provided that τ >1/2 and the wealth constraint is not binding. We continue with the case where the wealth constraint is not binding, but focusing now on the scenario where τ <1/2. Starting again from (B.23), we first notice that 2 q τ (A−C)− q 2(A−C)>−1, (B.27) since the left-hand side is monotone in τ and, for τ →0, 2 p τ (A−C)− p 2(A−C)→ − p 2(A−C), with− p 2(A−C)>−1 being implied by the assumption that 0≤A−C ≤ 1/2. Therefore, if τ > 1/2, 1+ h 2 p τ(A−C)− p 2(A−C) i ∈ [0,1] and, accordingly, two cases are possible: for a sufficiently high θ, 1>θ >1+ 2 q τ (A−C)− q 2(A−C) , (B.28) 177 condition (B.23) is satisfied, which is equivalent to p IP >p CP , whereas for a sufficiently low realization, 0<θ <1+ 2 q τ (A−C)− q 2(A−C) , (B.29) the ordering is reversed to p CP >p IP . Sincein thehigh uncertainty case condition (B.22) is trivially satisfied -implying p PI > p IP - for τ <1/2 and a sufficiently high realization of θ, the price ordering is p PI >p IP > p CP , whereas for a sufficiently low level of fundamentals we have both p CP > p IP and p PI >p IP , meaning that to get the final ordering we still need to compare p CP to p PI . Forthat, proceedingexactlylikein(B.24), for1>θ >1+A−C− p 2(A−C),itfollows that p PI >p CP , whereas if 0<θ <1+A−C− p 2(A−C), the ordering is p CP >p PI . In the high uncertainty case we are analysing, τ > A−C and, in particular, τ > (A−C)/4, which in turn implies that 1+A−C− q 2(A−C)<1+2 q τ(A−C)− q 2(A−C), (B.30) and, given the above, the ordering of prices for a state of fundamentals such that 0<θ <1+A−C− q 2(A−C) (B.31) is p CP >p PI >p IP , whereas if 1+A−C− q 2(A−C)<θ <1+2 q τ(A−C)− q 2(A−C), (B.32) the ordering that holds is p PI >p CP >p IP , completing the proof. 178 B.2 Equilibrium Conditions in the Imperfect Information Case without Government Intervention In the case of no government intervention, from (2.7) it is known that, for any i ∈ I, E e θ| ξ i = ξ i , and from (2.12) that agents opt to buy only if ξ i ≥ p+t, in case they can afford to invest in the asset, i.e., A≥p. The market clearing condition is given by 1 2τ Z θ+τ p+t dξ =K ⇔ θ+τ−p−t=2τK ⇔ p=θ+τ(1−2K)−t≡p NG . To support p NG as the equilibrium price, the following conditions must be satisfied 1. θ−τ ≤p NG +t≤θ+τ (integral condition); 2. p NG >0 (price positeveness); and 3. A≥p NG (feasibility). For the integral condition, θ−τ ≤p NG +t ⇔ K ≤1, (B.33) and 179 p NG +t≤θ+τ ⇔ 0≤K, (B.34) and therefore for 0≤K≤1 that condition is satisfied. For price positeveness p NG >0 ⇔ θ >−τ(1−2K)+t, (B.35) and finally, for feasibility we need A≥p NG ⇔ A−τ(1−2K)+t≥θ. (B.36) Combining the above yields the following Proposition B.3. For 0≤K ≤1, we have that, for any θ∈(−τ(1−2K)+t,A−τ(1−2K)+t]∩[0,1], (B.37) the following holds: 1. θ−τ ≤p NG +t≤θ+τ (integral condition); 2. p NG >0 (price positeveness); and 180 3. A≥p NG (feasibility), where p NG is the equilibrium price, given by p NG =θ+τ (1−2K)−t. (B.38) As a corollary we have Corollary B.1. Supposing true the conditions in the proposition above, the equilibrium is characterized by 1. X i =1, ∀i∈I :ξ i ≥θ+τ(1−2K), and X i =0 otherwise; 2. p NG =θ+τ(1−2K)−t. B.3 ComparisonofEquilibriumPricesacrossDifferentFrame- works without Government Intervention Proposition B.4. Given a unitary supply level, K = 1, the following ordering of equilib- rium prices prevails across the different informational scenarios - without the participation of the government: (i) If the wealth constraint is binding, the ordering of prices is p NGC >p NGP >p NG ; (B.39) (ii) If the wealth constraint is not binding, (a) A sufficiently low realization of the state of fundamentals, 0≤θ <1/2, (B.40) 181 implies the ordering of equilibrium prices to be p NGC >p NGP >p NG , (B.41) (b) A sufficiently high realization, 1/2 <θ≤1, (B.42) implies that ordering to be p NGP >p NG >p NGC (B.43) if 1/2+τ <θ≤1, and p NGP >p NGC >p NG , (B.44) in case 1/2 <θ <1/2+τ, with the equilibrium prices across the different scenarios given by (i) Imperfect information: p NG =θ−τ−t; (ii) Perfect information: p NGP =θ−t; and (iii) Common prior 3 : p NGC =A or p NGC =1/2−t. Proof. If the wealth constraint is binding, in the common prior case the equilibrium price is p NGC = A and - since feasibility requires the wealth being greater th an the price in any scenario - it follows that p NGC > max n p NG ,p NGP o . Given that τ > 0, p NGP > p NG and, therefore, the ordering of equilibrium prices is p NGC >p NGP >p NG . 3 In the common prior case, p NGC =A if the wealth constraint is binding. 182 If the wealth constraint does not bind, the equilibrium price in the common prior case is p NGC =1/2−t and, since it is still true that p NGP >p NG , it remains to compare p NGC to p NGP and to p NG , if need be. For that, p NGP >p NGC ⇔ θ >1/2 (B.45) and, therefore, according to condition (B.45), if 0≤θ < 1/2, then p NGP <p NGC , with the final price ordering being p NGC > p NGP > p NG . Otherwise, i.e., if 1/2 < θ ≤ 1, then p NGP > p NGC and, since p NGP > p NG , it follows that p NGP > max n p NG ,p NGC o , so that to obtain the final ordering of prices it is still needed to compare p NG to p NGC . In that regard, p NG >p NGC ⇔ θ >1/2+τ, (B.46) so that, following condition (B.46), p NGP > p NG > p NGC in case the level of funda- mentals is such that 1/2+τ < θ ≤ 1, with p NGP > p NGC > p NGP whenever 1/2 < θ < 1/2+τ. 183 B.4 Government vs No Government Prices in the Common Prior Scenario Proposition B.5. In the common prior scenario, the equilibrium price that prevails in the framework with the possibility of government intervention, p CP , is at least as high as the one that prevails in the framework without, p NGC , with the equilibrium prices being given by (i) Government: p CP =A or p CP = A−C K −t+1− r 2 A−C K ; and (ii) No Government: p NGC =A or p NGC =1/2−t. Proof. Recall that, when there is the possibility of government intervention, for the case of common priors about the fundamentals, the equilibrium price is either p CP =A, (B.47) if A∈ 0,(A−C)/K−t+1− p 2(A−C)/K , else p CP = A−C K −t+1− s 2 A−C K , (B.48) with (A−C)/K <1/2. For the case where government is absent, we have the equilib- rium price being either p NGC =A, (B.49) if A∈(0,1/2−t), else p NGC =1/2−t. (B.50) 184 We have, for 0<(A−C)/K <1/2, A−C K −t+1− s 2 A−C K > 1 2 −t, (B.51) which implies that, (i) For 0<A<1/2−t, the two equilibrium prices are the same, p CP =p NGC =A; (ii) For 1/2−t<A<(A−C)/K−t+1− p 2(A−C)/K, the equilibrium price in the casewherethereisthepossibility ofgovernmentintervention, p CP =A, ishigherthan the one when government is absent, p NGC =1/2−t; and (iii) For (A−C)/K −t + 1− p 2(A−C)/K < A, the equilibrium price with no gov- ernment is p NGC = 1/2−t, whereas with government, p CP = (A−C)/K−t+1− p 2(A−C)/K and, from (B.51), the later is greater than the former. Hence,unlessthewealthofinvestorsislowenoughsothatp CP =p NGC =A,theequilib- rium price when agents entertain the possibility of government intervention is undoubtedly higher than the one that prevails when government is absent. 185

Abstract (if available)

Abstract This thesis studies the effects of government intervention in financial crises. In the first chapter it is analysed the effects of intervention for the formation of the network of banks, whereas in chapters 2 and 3 it is analysed the effects of intervention on asset prices. ❧ Chapter is divided in two parts. In the first one, government intervention is defined as a policy that allows banks to reduce fire sale costs, and an endogenous model of the formation of the network of banks is developed in a way that this type of intervention plays a key role in determining the structure of the banking system. The idea is that intervention brings incentives for banks to invest in more profitable but less liquid assets, which in turn can be financed only through interbank loans, which creates links across banks. In the second part of the paper, financial fragility is defined as the number of bank failures after banks' assets are hit by shocks, and under this measure networks obtained with and without intervention are compared. Theoretically it is shown that intervention makes the network of banks to be more connected, and from simulations that it brings more financial fragility despite increasing the wealth of the banking system. ❧ Chapter 2 studies the effect of a policy whereby intervention occurs only in financial crises where the welfare of investors goes below a specific threshold, to be called the safety net. The main question is how equilibrium prices differ under different intervention policies, and for that it is studied the investment decision problem of agents under two different frameworks, with and without intervention, and under three different informational scenarios, imperfect information, perfect information, and common priors. These informational scenarios are meant to capture different classes of assets where investors have varying degrees of knowledge regarding the technology embedded in the asset. It is shown that, regardless of the informational scenario, equilibrium prices can be sustained at a higher level in the framework with intervention than in the one without. However, equilibrium prices cannot be sustained at too much of high levels because, even though that would signal to investors that intervention would take place for sure, it still would not make investments profitable and, therefore, agents would prefer not to participate in the market, making the market clearing condition to fail to hold and an equilibrium not to exist. ❧ Chapter 3 studies the effects of a policy such that intervention occurs only in those financial crises where there is a significant drop in the level of prices. It is assumed that everyone knows the observed price of a particular asset, but only the government knows the fundamental price, the difference between the two being called a bubble. Nature defines if there will be a crisis or not and, every time there is one, the bubble bursts and the observed price drops to the fundamental. At this moment the government faces the cost of letting the bubble burst - proportional to the size of the bubble - and the cost of intervention, and by trading off the two it decides on its action. It is shown that, under such a policy rule, investors have incentives to inflate bubbles because by doing so they can make a higher capital gain if there is no crisis and, in case there is one, they increase the likelihood of intervention.

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Creator Lopomo Beteto, Danilo (author)

Core Title Essays on government intervention in financial crises

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Economics

Publication Date 05/30/2013

Defense Date 02/26/2013

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag bubbles,financial crises,fragility,government intervention,government safety net,networks,OAI-PMH Harvest

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Magill, Michael (committee chair), Quadrini, Vincenzo (committee member), Wang, Yongxiang (committee member), Zapatero, Fernando (committee member)

Creator Email beteto@usc.edu,danilobeteto@yahoo.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-257936

Unique identifier UC11294057

Identifier etd-LopomoBete-1704.pdf (filename),usctheses-c3-257936 (legacy record id)

Legacy Identifier etd-LopomoBete-1704.pdf

Dmrecord 257936

Document Type Dissertation

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Rights Lopomo Beteto, Danilo

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Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

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