Revision Lecture. MSc Finance: Theory of Finance I MSc Economics: Financial Economics I

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1 Revision Lecture Topics in Banking and Market Microstructure MSc Finance: Theory of Finance I MSc Economics: Financial Economics I April 2006 PREPARING FOR THE EXAM What do you need to know? All the models we studied in the lectures. How should you prepare for the exam? 1. Use the texts and handouts to review the material. 2. Try to summarise each model with and without formal algebra. 3. Can you solve through a model using special cases and /or numerical values? Any tips for the examination? 1. Read the questions carefully. Sometimes the material sounds familiar but differs slightly from the class material. 2. Pay special attention to notation: it may not be the same as in handouts. 3. Answer questions as completely as possible. Partial answers get little credit. 4. Explain your answers carefully. You can be brief AND precise. 1

2 1 Topics in Banking 1.1 Inter-temporal Optimization Can you analyse a simple, two-period, model of inter-temporal consumption choice? In the handout for lecture 1, review the inter-temporal budget constraint, the Lagrangean and the conditions for the optimal consumption profile. Can you solve the model for standard utility functions, such as logarithmic utility? See exercises 1 and 2 in the handout for lecture Expected utility Can you define expected utility and compute it in simple contexts? Do you recall Jensen s inequality? What is the link between risk aversion and curvature (concavity/convexity) of the utility function? How do we measures of risk aversion (absolute risk-aversion, relative risk aversion)? Can you solve for the optimal portfolio in simple contexts, with given utility functions? See exercises 3 and 4 in the handout for lecture 1. Demonstrate that a risk averse who is offered insurance at acturially fair rates will buy complete insurance. 1.3 Liquidity Insurance & Bank Runs Review the material in the handout for lecture 2 and 6. Can you provide a brief summary of the Bryant /Diamond-Dybvig models of liquidity insurance and bank runs? You should be able to outline the basic structure what kind of shocks do individuals face, the investment technology, their optimal choices in various environments (in autarky, with securities or bond markets, with fractional banking), and be able to rank these by in terms of efficiency. 2

3 Define a bank run. Explain why fractional reserve banking is vulnerable to bank runs. You should be able to demonstrate understanding of the bigger picture. In this case, the idea is that financial markets and institutions provide pools of liquidity. This liquidity is valuable when individuals face idiosyncratic liquidity shocks. Liquidity provision may make is easier to finance investment in illquid assets, increasing welfare. Fractional reserve banking is one way to implement this. RECENT EXAM QUESTIONS ON THIS TOPIC (June 2004) Derive the Diamond-Dybvig model of banks as providers of liquidity insurance. Describe in detail the market allocation (when real investments are securitized and traded in an intermediate stock market), and the allocation with intermediaries (when real investments are carried out by a bank that is financed by deposits from consumers/investors). ANSWER: This requires a careful summary of the model. (June 2005) Consider a three-period economy with a continuum of agents. Each agent is endowed with one unit of a good in period 0, which she must use to finance consumption c 1, c 2 in periods 1 and 2, respectively Agents have access to a storage technology that allows costless transfer of the good from one period to another. They can also invest in a two-period productive technology: each unit invested in this technology in period 0 returns R > 1 in period 2 while premature liquidation in period 1 returns L < 1. The agents are ex-ante identical but subject to consumption uncertainty: with probability π an agent will want to consume only in period 1, and with probability 1 π she will want to consume only in period 2. The ex ante utility function of all agents is U = πu(c 1 ) + β(1 π)u(c 2 ), where β < 1 is a discount factor. Assume that βr > 1. (a) What is the outcome if agents cannot trade with each other (i.e., in autarky)? Explain why this outcome is inefficient. 3

4 ANSWER: In the absence of trade, each agent much choose how much to invest in the two-period investment technology and how much to keep in the more liquid storage technology. If she invests an amount I and stores the rest, 1 I, the consumption possibilities are c 1 = LI + 1 I, if she has to liquidate and consume early, and c 2 = RI + 1 I < R, if she does not liquidate early. The agent will choose I to maximise expected utility subject to these constraints. Even without solving for this, it is easy to see that the outcome is ex-post inefficient. (Here c 1 < 1 and c 2 < R as long as 0 < I < 1. In the event of preference for early consumption, the agent would have preferred zero investment in the productive technology and consumed 1; in the event of preference for later consumption, she would have preferred to invest everything in the productive technology and consumed R. The actual consumptions levels are strictly lower in each event. Hence the inefficiency. (b) Obtain a condition for the optimal symmetric allocation in this economy. ANSWER: The optimal symmetric allocation maximizes subject to πu(c 1 ) + β(1 π)u(c 2 ) πc 1 = 1 I (1 π)c 2 = RI The first constraint requires that the aggregate consumption of the early consumption types is met through storage. The second constraint requires that the consumption of the patient types be met from the proceeds of the productive technology. Substituting from the constraints, we can write the objective function in terms of I alone: πu( 1 I RI ) + β(1 π)u( π 1 π ), with first-order condition u (c 1 ) + βru (c 2) = 0. 4

5 (c) Explain how a system of fractional reserve banking can implement the optimal outcome. Identify any assumptions. ANSWER: Under a fractional reserve system, the bank collects deposits from all agents: it invests a fraction in the two-term productive asset and stores the rest. A deposit contract specifies the amounts (c 1, c 2 ) that can be withdrawn at various periods for 1 unit deposited at time 0. Typically this outcome is sustainable only when βr > 1: if not, we have c1 > c 2, so that even patient agents would like to withdraw in period 1 and store the proceeds till they consume it. (d) What is a bank run? Why is fractional reserve banking sometimes vulnerable to bank runs? ANSWER: A bank run refers to a situation in which depositors seek to withdraw their funds due to the fear that others withdrawals causing the bank to fail. Such situations may arise as Nash equilibria, so are entirely rational from an individual point of view. Details of the various possibilities are as in the handout. (Christmas 2005) Consider a three-period economy with a continuum of agents. Each agent is endowed with one unit of a good in period 0, which she must use to finance consumption c 1, c 2 in periods 1 and 2. Agents have access to two technologies. Technology α is a one-period technology that returns A for every unit invested in the previous period. Technology β is a two-period technology: each unit invested in this technology returns B after two periods, while premature liquidation after one period returns nothing. The agents are ex-ante identical but subject to consumption uncertainty: with probability π an agent will want to consume only in period 1, and with probability 1 π she will want to consume only in period 2. The ex ante utility function of all agents is U = π ln(c 1 ) + δ(1 π) ln(c 2 ), where δ < 1 is a discount factor. Assume that δb > 1. Note that this is a slightly altered version of the standard model we covered in class. The classroom version had different notation, assumed A = 1, and premature liquidation yielded L > 0. These minor changes are designed to test your understanding. 5

6 (a) What is the autarkic outcome (that is, if agents cannot trade with each other)? Identify any assumptions you make. Explain why this outcome is inefficient. ANSWER: In the absence of trade, each agent invest in each technology. If she invests an amount I in technology β and 1 I in α, the consumption possibilities are c 1 = A(1 I), if she consumes early, and c 2 = BI + A 2 (1 I), if she consumes late. The agent will choose I to maximise expected utility. π ln(a(1 I)) + δ(1 π) ln(bi + A 2 (1 I)) The first-order condition for an interior maximum is Aπ A(1 I) + δ(1 π)(b A2 ) BI + A 2 (1 I) assuming B > A 2. [Note that if B A 2, it is optimal to set I = 0. Investment in = 0, technology α has both a higher return and greater liquidity.] We solve this to get and then solve for I = (B A2 )(1 π)δ πa 2 (B A 2 )[π + (1 π)δ], c 1 = A(1 I ) c 2 = BI + A 2 (1 I ) We can see that this is inefficient even without solving explicitly,. (In the event of preference for early consumption, the agent would have preferred all investment in technology α and consumed A; in the event of preference for later consumption, she would have preferred to invest everything in the technology β and consumed B. The actual consumptions levels are strictly lower in each event, so this outcome is inefficient. (b) Specialise the analysis to the case where A = 1. Obtain the optimal symmetric allocation in this economy. ANSWER: Set A = 1. Then c 1 = 1 I c 2 = BI + (1 I) 6

7 The optimal symmetric allocation maximizes π ln(c 1 ) + δ(1 π) ln(c 2 ) subject to πc 1 + (1 π) c 2 B = 1. The first-order condition requires c 2 = δbc 1 Substituting this in the budget constraint requires πc 1 + (1 π)δc 1 = 1 or and c 1 = 1 π + (1 π)δ c 2 = δb π + (1 π)δ. (c) Explain how a system of fractional reserve banking can implement the optimal outcome. Identify any assumptions. ANSWER: Under a fractional reserve system, the bank collects deposits from all agents and invests appropriate amounts. A deposit contract specifies the amounts (c 1, c 2 ) that can be withdrawn at various periods for 1 unit deposited at time 0. Typically this outcome is sustainable only when δb > 1: if not, we have c1 > c 2, so that even patient agents would like to withdraw in period 1 and reinvest in technology α. (d) What is a bank run? Why is fractional reserve banking vulnerable to bank runs? ANSWER: A bank run refers to a situation in which depositors seek to withdraw their funds due to the fear that others withdrawals causing the bank to fail. Such situations may arise as Nash equilibria, so are entirely rational from an individual point of view but inefficient. See the handout for details 1.4 Debt contracts and Risk Sharing Optimal Risk Sharing with Symmetric Information Review Section 2.1 in handout for lecture 6. Explain why standard debt contracts do not imply optimal risk-sharing arrangements when lenders (large banks) are less risk- 7

8 averse than borrowers and all information is symmetric. Efficient Incentive-Compatible Contracts with Asymmetric Information Review Section 2.2 in handout for lecture 6. Summarise the basic model with asymmetric information and costly state verification. Can you explain why the standard debt contract is the efficient, incentive-compatible contract? 1.5 Moral Hazard in Debt Markets Incentives to Repay: The Threat of Termination Review Section 2.3 in handout for lecture 6. Outline a two-period model that shows how the threat of termination of future lending can create incentives to repay debt. Sovereign Debt Review Section 2.4 in handout for lecture 6. Assume, in an infinite horizon model, that a sovereign borrower who defaults will not be able to borrow again. If the return to borrowing, given by the function V(L), has decreasing returns to scale, show that the borrower faces a credit limit (i.e., there exists some L such that borrowing must be less than L to preserve the incentive for repayment. Moral Hazard, Direct and Intermediated Lending, 1.6 Adverse Selection What is adverse selection? How does it affect debt markets? Review Section 2.5 in handout for lecture 6. Outline a simple model to show how collateral can solve the adverse selection problem Review Section 1 in handout for lecture 7 (this is the Leland-Pyle model). Outline a model to how partial self-financing can serve as a signalling device. 8

9 Review Section 2 in handout for lecture 7. What is credit rationing? Explain the central argument of the Stigltiz-Weiss model. EXAM QUESTION (2002) Suppose the return of an investment project, R(θ), is normally distributed with mean θ and variance σ 2. The project costs 1. An entrepreneur has mean-variance preferences represented by the utility function u. The absolute risk aversion coefficient is ρ > 0, the investor has initial wealth W and, therefore, end of period wealth W 0 + R(θ) after investment. The expected utility is Eu(W 0 + R(θ)) = u(w 0 + θ ρ 2 σ2 ) Although the entrepreneur has sufficient funds to self-finance the project, he can also seek outside financing from a risk-neutral investor. (a) What potential gains can you see from external financing of the project? ANSWER: External financing can allow the risk-averse entrepreneur to transfer risk (which he dislikes) to the risk neutral investor (who does not mind the risk). (b) If the entrepreneur can sell the project to an outside investor, what is the minimum price that he would accept and what is the maximum price the investor would pay? ANSWER: If the entrepreneur finances the project himself, his expected utility is Eu(W 0 + R(θ)) = u(w 0 + θ ρ 2 σ2 ). If he sells the project at price P, he trades the uncertain return R(θ) for the certain value P. The expected utility after sale is Eu(W 0 + P) = u(w 0 + P). The entrepreneur will be willing to sell as long as the utility after sale exceeds the utility prior to sale, that is as long as u(w 0 + P) u(w 0 + θ ρ 2 σ2 ), or that W 0 + P W 0 + θ ρ 2 σ2, 9

10 or that P θ ρ 2 σ2. Hence the minimum price at which the entrepreneur would be willing to sell is θ ρ 2 σ2. The investor is risk neutral, so values each project by its mean return. Since the project R(θ) has expected return θ, that is the maximum he is willing to pay. (c) Now suppose the investor can observe the variance of the project (σ 2 ) but not the expected return (θ), so the price cannot depend on θ. Assume that the investor knows the probability distribution of θ, where θ is high (θ 1 ) or low (θ 2 ) with probability π 1 and π 2 respectively. Outline the condition for when it is optimal for the entrepreneur to seek outside financing. When is the investor more likely to self-finance the project? Explain your answer. ANSWER: Note the change in notation in this exam question from that in the textbook. You need to be careful about notation in exam questions. If an investor self finances the project, he gets expected utility Eu(W 0 + R(θ i )) = u(w 0 + θ i ρ 2 σ2 ), where he will know if θ i = θ 1 (high) or θ i = θ 2 (low). If he can sell the project for a fixed price P, he gets Eu(W 0 + P) = u(w 0 + P). Selling is a good idea as long as P > θ i ρ 2 σ2. It follows directly that entrepreneurs who know their project to be bad (θ 2 ) will be more inclined to sell. (d) Leland and Pyle s (1977) paper describes a signalling equilibrium for this situation. In this signalling equilibrium, what is being signalled, and how? ANSWER: In the equilibrium, the quality of the project (that is, its expected return) is being signalled, by the entrepreneur being willing to self-finance a part of the project. (e) Derive the Leland and Pyle equilibrium algebraically ANSWER: Reproduce the equilibrium described above, with the no-mimicking condition. Recall that you need to adapt the notation. In the signalling equilibrium entrepreneurs of good project will self finance at least a proportion α of their project and sell the rest at a unit price P 1 = θ 1. Entrepreneurs 10

11 with bad projects will sell their entire project for a lower unit price P 2 = θ 2. To ensure that entrepreneurs with bad projects do not want to mimic entrepreneurs with good projects, they must get a higher utility by selling their entire project at the lower price u(w 0 + θ 2 ) Eu(W 0 + α R(θ 2 ) + (1 α)θ 1 ) = u(w 0 + αθ 2 ρ 2 α2 σ 2 + (1 α)θ 1 ), or equivalently θ 2 αθ ρσ2 α 2 + (1 α)θ 1, or α 2 1 α 2(θ 1 θ 2 ) ρσ 2. EXAM QUESTION (2005) Consider a population of investors, each with a project that has two possible outcomes: a positive return y in the event of success, a zero return in the event of failure. Projects differ in their risk of failure: if θ denotes the probability of failure, a fraction π of projects have high risk of failure θ H, while the rest have lower risk of failure θ L < θ H. Investors have no resources of their own, so must borrow to invest and must repay R if the project succeeds. The lender cannot observe the probability of failure of a project directly but knows the fraction of each type in the population. (a) Develop the above structure to explain how the market for loans may be vulnerable to adverse selection. Note any assumptions you make. ANSWER Adverse selection: If the lender could observe the risk type, it would offer separate contract for each type to reflect their risk characteristics. Assume that lenders have the bargaining power but cannot squeeze borrowing investors below their reservation utility. If U k denotes reservation utility of risk-type k, the highest sustainable interest rate must satisfy (1 θ k )(y R k ) = U k, so that high-risk types pay R H = y UH 1 θ H 11

12 while low-risk types pay We assume that U L 1 θ L R L = y > UH 1 θ H so that R H > R L. UL 1 θ L. If the lender cannot observe the risk types it must offer the same contract to each. If it finds it profitable to lend to each type, it must offer a contract that is just sweet enough for the low risk type (i.e set R = R L ): if so, high-risk types would extract an informational rent. If lending to low risk types is not profitable, only the high risk types borrow and pay R H. (b) How can collateral requirements reduce the problem of adverse selection? ANSWER: Collateral, C: something that the borrow loses in the event of failure. Often inefficient, in that the lender recovers only some fraction of the value. That is recovers δc, with δ < 1 The lender could offer two contracts (C H, R H ) and (C L, R L ) such that each type selfselects into the right type. It turns out that given that collateral is inefficient, the first contract will a high interest rate but no collateral requirement (so that C H = 0); the second contract has lower interest rate with a collateral requirement C L > 0. High risk types will select the contract without collateral (1 θ H )(y R H ) (1 θ H )(y R) θ H C L Low risk types will select the contract with collateral (1 θ L )(y R) θ L C L U L The role of collateral here is to allow for self selection. 12

13 2 Topics in Market Microstructure 2.1 Trading models The chapter references are from Market Microstructure Theory by Maureen O Hara, Blackwell, Information Based Models: Sequential trading - Glosten-Milgrom (1985) Chapter 3 - Sections 3.1, 3.2, 3.3, 3.5 and the Bayesian updating problem described in the appendix. Strategic Trading - Kyle (1985): Chapter 4 - section 4.1.1, handout 2.2 Herd Behaviour The chapter references are from Rational Herds, by Christophe P. Chamley, Cambridge University Press, Chapter 2 - up to & including (Basics) Cascades and Herds: Chapter 4 - up to & including proposition 4.3, handout Gaussian model: Chapter 3 - up to the end of the first subsection ( An intuitive description... ) of section Auction Theory Handout on auctions - contains a description of basic single-unit auction theory, basic multi-unit auctions, and some applications. You should study the handout carefully. In particular, the following topics from the handout are important. You are expected to know about these topics only to the extent covered in the handout. Revenue equivalence Optimal bid in Vickrey auction 13

14 Winner s curse Revenue ranking under common values Shill bidding Sniping Proxy bidding Multi unit auctions: Uniform price auctions - example to show that equilibrium need not involve truthful bidding The multi-unit Vickrey auction Auctions versus beauty contests For MSc Economics (Option) only: In addition to this overview, you need to read the following. Auctions: Theory and Practice by Paul Klemperer, Princeton University Press, 2004: Chapter 1 (overview only); Chapter 3 - practical concerns in auction design; Chapters 5 and 6 - Telecom auctions (UK and across Europe) 14

15 2.4 Answers to some past questions Kyle model 1. Final Exam This question is based on the Kyle (1985) model. The value of an asset being traded is a random variable: V N(0, σ0 2 ). There is an informed trader who gets an informative signal on V. We assume the signal is perfect, so the informed trader knows the realization of V. There are also noise traders who trade for exogenous reasons and submit a trade quantity u which is a random variable: u N(0, σ 2 u). Here u and V are independent random variables. After observing V, the informed trader chooses his trade quantity X. The market maker observes the net order flow (X + u) and sets a price P(X + u) so that he breaks even (earns a zero expected profit). Thus P(X + u) = E(V X + u). Show that there exists a linear equilibrium of the form X(V) = βv P(X + u) = λ(x + u) by explicitly deriving the values of the constants β and λ. You can make use of the following property of a Normal distribution. Suppose a random variable θ N(m 0, σ 2 0 ). Consider a signal s = θ + ɛ, where ɛ N(0, σ2 ɛ ) (and ɛ is independent of θ). Then the posterior distribution of θ is given by N(m 1, σ 2 1 ) where m 1 = αs + (1 α)m 0 σ 2 1 = σ2 0 σ2 ɛ σ σ2 ɛ where α = σ2 0 σ σ2 ɛ 15

16 ANSWER Let X(V) = βv. We need to calculate E(V X + u). Now, X + u = βv + u. Let Z = X + u. β Note that E(V X + u) = E(V Z). Further, Z = V + u β, where u β N(0, σ2 u β 2 ). Then E(V Z) = αz + (1 α)0 where α = σ 2 0 σ (σ2 u/β 2 ). Rewriting using the value of Z, the market makers price, given by the posterior expectation, is P(X + u) = E(V X + u) = α (X + u). (1) β The expected profit of the informed trader is E(π V) = E[(V P(X + u))x V] = (V E(P(X + u)))x = (V α β X)X, where the last step follows from the fact that Eu = 0. Maximizing the expected profit, we get the optimal trade as X = β 2α V We started by assuming X = βv. For this to be correct, we need to have β 2α = β which implies α = 1/2. Thus σ 2 0 σ (σ2 u/β 2 ) = 1 2. From this we get β = σ u σ 0. Thus indeed, if the market maker sets price to equal posterior expectation, a best response of the informed trader is to submit a trade that is a linear function of his information, where the linear function is as stated in the result. Further, substituting the values of α and β in equation (1), we get P(X + u) = 1 2 σ 0 σ u (X + u). Thus the market maker s price is linear in his signal, and the coefficient λ is given by 1 σ 0. 2 σ u 16

17 2. Final Exam Consider a risky asset with final payoff Ṽ, which is a normally distributed random variable with mean 0 and variance σ0 2. There is a risk neutral informed trader who knows the realization v of the random variable Ṽ. The trader has no initial endowment of the risky asset. After observing v, the informed trader chooses his trading quantity X. This net order flow consists of the informed trader s order and the total order from liquidity traders (denoted by u) which is normally distributed with mean 0 and variance σ 2 u. All random variables are pairwise independent. The market maker observes the net order flow X + u and sets a price P(X + u) = E(V X + u). Consider the linear equilibrium given by and X(v) = σ u σ 0 v, P(X + u) = σ 0 2σ u (X + u). (a) Show that if liquidity trade quantity doubles, P(X + u) remains unchanged in equilibrium. Explain the intuition behind this result. (b) Derive the expected profit of the informed trader conditional on his information, and show that it is decreasing in σ 0. Explain the intuition behind this result. (c) Derive the expected ex-ante profit of the informed trader, and show that it is increasing in σ 0. Explain the intuition behind this result. ANSWER (a) P(X + u) = σ 0 (X + u) = σ ( ) 0 σu v + u = v 2σ u 2σ u σ σ 0 2 u σ u If u doubles, so does σ u - so P is unchanged. Intuition: if liquidity trade quantity doubles, the order flow of the informed trader doubles as well, as he can now better hide behind the uninformed order flow. 17

18 (b) using Eu = 0. E(π v) = E(X(v P) v) = σ u σ 0 v(v v 2 ), Thus which is decreasing in σ 0. E(π v) = σ u σ 0 v 2 2, Intuition: Rise in σ 0 lowers ability of informed bidder to hide his trade behind noise trade. (c) which is increasing in σ 0. E(π) = σ u σ 0 Ev 2 2 = σ u σ 0 σ = σ uσ 0 2, Intuition: A rise in σ 0 has the effect described in (b), but also has the additional effect ex-ante of making higher absolute values of v more likely, which increases profit. The calculation shows that this second effect dominates. 18

19 2.4.2 Herd behaviour Final Exam Consider the following version of the model of Bikhchandani, Hirshleifer and Welch (1992). A random variable θ has two possible values, 0 and 1. At the start of time, the prior is Prob(θ = 1) = 1/2. There are N > 2 agents indexed by t 1, 2,..., N. Agent t receives a private signal s t {0, 1} which signals the value of θ with precision 3/4, i.e. Prob(s t = θ θ) = 3 4. Agents take action sequentially, with agent t acting in period t. Each agent chooses an action x from the discrete set {0, 1}. Action x = 0 is costless while the cost of action x = 1 is 1/3. The payoff of any agent is given by: { 0 if x = 0 u(x, θ) = θ 3 1 if x = 1 Before deciding on x, agent t knows his own signal and the history of actions chosen by agents who act before agent t. Agents are risk neutral and choose the x that maximizes expected payoff. If both actions yield the same expected payoff, agents choose x = 0. (a) Calculate the optimal action of agent 1 for each of the two signals. Answer If agent 1 receives signal s 1 = 1, the posterior is Prob(θ = 1 s 1 = 1) = = Prob(θ = 1) Prob(s = 1 θ = 1) i Prob(θ = i) Prob(s = 1 θ = i) (1/2)(3/4) (1/2)(3/4) + (1/2)(1/4) = 3 4 Similarly, the posterior for signal s 1 = 0 is Prob(θ = 1 s 1 = 0) = 1/4. For signal s 1 = 1, x = 1 yields the expected payoff Eθ (1/3) = (3/4) (1/3) = (5/12) > 0. Thus for s 1 = 1, the optimal action is x = 1. For signal s 1 = 0, x = 1 yields the expected payoff Eθ (1/3) = (1/4) (1/3) = (1/12) < 0. Thus for s 1 = 0, the optimal action is x = 0. Thus agent 1 takes into account both the public signal and private signal in deciding on optimal action. 19

20 (b) Suppose agent 1 receives signal 1 and acts accordingly. Show that an information cascade starts from the next period. Answer From above, s 1 = 1 implies x = 1. Now suppose agent 2 receives a signal s 2 = 1. Clearly, his optimal action is x = 1 (the posterior is even higher than 3/4). Suppose s 2 = 0. The new prior is 3/4. The posterior is Prob(θ = 1 s 2 = 0) = = Prob(θ = 1) Prob(s = 0 θ = 1) i Prob(θ = i) Prob(s = 0 θ = i) (3/4)(1/4) (3/4)(1/4) + (1/4)(3/4) = 1 2 Now, x = 1 yields the expected payoff Eθ (1/3) = (1/2) (1/3) = (1/6) > 0. Thus for s 2 = 0, the optimal action is x = 1. Agent 2 takes the action x = 1 irrespective of his private information. Thus subsequent agents learn nothing new and adopt x = 1. Thus a cascade starts from period 2. (c) Briefly describe the difference between an information cascade and herd behaviour. Outline Answer: Cascade : agents ignore own information. Herd: agents optimally choose (after taking private information into account) to take the same action. Cascade: Arises only in special cases in which the action space and the signal space are very coarse (you should clarify this with a few sentences about the BHW model). Thus a cascade is not a robust phenomenon. A cascade is also fragile (can be reversed by revealing some public information). Herd: Robust, and not fragile. Cascade: It is possible that once a cascade starts, all subsequent agents take an inefficient action. No social learning. Herd: with rich enough signal space (e.g. Gaussian model), social learning is efficient. In this case the only problem could be the speed of learning if signal extraction is imperfect because of noise. 20