Ec2723, Fall 2013: Assignment 4

Optional review questions to prepare for the nal exam.

Answers will be posted during the reading period.

1. A literature similar to the one discussed in class (Alvarez-Jermann 2000, and alsoKehoe-Levine 1993 and Kocherlakota 1996) has been developed in international economicsin order to understand the sustainability of sovereign debt. The classic papers in thatliterature are Eaton-Gersovitz (1981), which shows that even when the possibility of defaultexists, the punishment of autarky from nancial markets is enough to sustain a positivelevel of debt and prevent default, and Bulow-Rogo (1989), which shows that allowing forlending (but not borrowing) after default makes the amount of sustainable debt equal tozero. Hellwig-Lorenzoni (2009) links both results and concludes that economies with lowinterest rates allow for self-enforcing debt in equilibrium even when lending is possible afterdefault. This result diers from Bulow-Rogo because interest rates are now endogenous,while Bulow-Rogo implicitly assume high interest rates.

There are two innitely lived consumers j = 1; 2 who receive endowments yj (st). st

denotes possible date-t states and s0 corresponds to the initial date-0 state. Both agentsmaximize expected utility E

P1t=0

tu (cj (st)). Each agent enters state st with net asset

positions aj (st). At state st, each agent chooses net asset positions aj (st+1) in one-periodArrow-Debreu securities that are subject to the exogenous borrowing constraint j (st+1),i.e., aj (st+1) j (st+1). The price at st of a one period Arrow-Debreu security that deliversone unit at state st+1 is q (st+1). The market clearing conditions in any equilibrium areP

j cj(st) =

Pj y

j(st) andP

j aj(st) = 0.

a) Write the optimization problem solved by each agent at time t, taking care to statethe objective function, the ow budget constraint, and the borrowing constraint.

Assume now that, in each period, one consumer receives endowment e and the otherreceives e < e, such that e + e = 1. The economy alternates between two states of nature.In state 1, consumer 1 has the high endowment and in state 2, consumer 2 is the one withthe high endowment. The probability of switching state is . Lets solve for a (symmetric)stationary equilibrium ch; cl; qc; qnc; ! and aj(st). ch is the consumption of the agent withthe high endowment and cl is the consumption of the agent with the low endowment. qc isthe price of the Arrow-Debreu security that pays when the state changes and qnc is the onethat pays when there is no change of state. ! is the borrowing limit, constant across agentsand securities in this equilibrium. aj(st) denotes the net asset holdings of agent j when heenters state st. Remember that these asset holdings are determined in the previous period:aj(st) > 0 implies that the agent bought Arrow-Debreu securities in period t 1 that payat state st.

Assume that the initial state is 1, so agent 1 has high endowment in the initial state, andthat the initial asset positions are a1 (s0) = ! and a2 (s0) = !.b) Write the expected lifetime utility of an agent who starts with a high endowment in

this economy, as a function of ; ; u (ch) ; and u (cl).

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c) Conjecture that agents choose asset positions that pay ! in high endowment statesand a ! in low endowment states (a will be determined soon in equilibrium). Knowingthis, write the budget constraints for high and low endowment states. Substituting for a,combine both constraints to nd an intertemporal budget constraint.

d) What does the intertemporal budget constraint look like in the space ch; cl? How doesits slope relate to the Arrow-Debreu security prices?

e) Write the rst order conditions and characterize ch as a function of qnc and qc. Assumech < e. What is the value of qnc in equilibrium?

f) Assume now that 1 = qc+ qnc. As we will see, this assumption guarantees that privatedebt in equilibrium is self-enforcing, which means that debt limits are not too tight inthe sense that V (in equilibrium fully constrained) = V (default with possibility of lending),with V () denoting value functions. What is the value of qc under this assumption? Isa = ! a consistent value for asset holdings in equilibrium? Use ch = c, cl = c to denotethe equilibrium values for consumption. Can you express ! as a function of e, c, and ?Characterize c as a function of and . Check that all these conjectures are consistent witha stationary equilibrium. Remember that you have to verify that rst order conditions hold,markets clear and budget constraints are satised.

g) Defaulting in this economy does not imply autarkic consumption as in Alvarez-Jermann(2000), since an agent who has defaulted is able to lend but not to borrow. Formallyj(st+1) = 0 for all future nodes. In our example, default would imply ! = 0. Note that ifan agent decides to default at state st, he keeps the st assets. Looking at the intertemporalbudget constraint, what condition relating qc and qnc makes default attractive? How doesthis condition relate to the level of interest rates? Give economic intuition for this condition.

h) Plot the budget constraint and the indierence curve that selects the equilibrium c, cin the space ch; cl. Remember that this equilibrium is the one with self-enforcing private debt(qc + qnc = 1). Assume that c < e. Argue that allocations in the intertemporal constraintto the upper left of c; c will induce default. Plot the indierence curve that goes throughthe endowment point. Denote with ca; ca the consumption bundle that would arise in anequilibrium with positive risk sharing but with the punishment of complete autarky (thesame one as in Alvarez-Jermann 2000). Do you observe more or less risk sharing? (Whenplotting, note that c < e and the fact that utility is convex makes the bundle c; c preferredto the endowment.) Give some intuition about this result.

i) In the autarkic allocation, what are the values for qanc and qac ? Note that the assumption

c < e implies that 1=(qanc + qac ) < 1, i.e., the autarkic interest rate is less than 1. Does the

economy in autarky have high or low implied interest rates according to the denitionin Alvarez-Jermann (2000)? What if c e? Does this result have any relation with thefact that if autarky has high implied interest rates there is no possible risk sharing? Canyou rationalize the fact that when default is punished by autarky the equilibrium of theeconomy has high implied interest rates, but when default is punished with no borrowingthe equilibrium features low implied interest rates?

j) Bonus: Can you relate these results to the literature on rational bubbles?

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2. Consider a two-period model in which a unit mass of traders choose their portfoliostoday and the assets pay o tomorrow. There is a riskless bond yielding a gross return Rwhich is in perfectly elastic supply. There are also two risky stocks denoted i = 1 and 2.The terminal value of stock i is vi where vi

iid N (m; 1=) (we call the inverse of the variance,here , the precision). The per capita supply of the stock i, denoted xi, is distributed as

xiiid N(x; 1=) . Agents trade today at prices (1; p1; p2), where the bond price is normalized

to one. All investors have CARA utility with coe cient of absolute risk aversion > 0.Investors have an initial endowment w0 that they can use to purchase securities.

Conditional on a random draw of vi, traders may receive a value-relevant signal sijvi N(vi; 1=) about each stock. All traders see signal s2 about stock 2; so it represents publicinformation. Signal s1 is seen by only fraction of the traders, so it corresponds to privateinformation. All signals are received before trade begins. With the exception of the relation-ship between si and vi, all of these random variables are independent and all traders knowtheir distributions.

a) First we solve for the equilibrium price, p2 (s2; x2), and for expected returns, E [v2 Rp2],for asset 2. Note that, due to the assumed independence of the relevant random variables,this problem is entirely separate from that of solving for prices and expected returns for asset1. Thus, asset 2 provides a symmetric information benchmark for asset 1.

(i) Since all agents see s2, all agents have the same beliefs about asset 2: Show that eachagents posterior is given by v2js2 N (v2 (s2) ; 1=2) where

v2 (s2) =m+ s2+

and 2 = + :

Since all agents have the same beliefs about asset 2, they have the same demands. Solve foreach agents demand for asset 2 as a function of the signal, s2, and the price p2:

(ii) Compute the market clearing price p2 (s2; x2) as a function of the signal, s2, and therandom supply, x2. Compute the expected return, E [v2 Rp2], on asset 2.b) Next we solve for the price and expected return for asset 1: We conjecture a linear

pricing function of the form

p1 (s1; x1) = a m+ b s1 c x1 + d x:

(i) Find the posterior beliefs of the informed conditional on observing s1 and p1. Find theasset demands of the informed, say xI1 (s1; p1). (Hint: Do the informed learn any additionalinformation about the nal payo, v1, from observing the price?).

(ii) We now nd the posterior beliefs and asset demands of the uninformed. Given ourconjectured pricing function, the uninformed eectively observe

1 =p1 a m+ (c d)x

b= s1 c

b(x1 x)

which is s1 plus noise. Note that the precision of 1, say , depends on the conjecturedpricing function through the ratio c=b. Based on this observation, nd the posterior beliefsand demands of the uninformed, say xU1 (p1).

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(iii) For a given random supply, x1, impose market clearing and solve for the equilibriumprice as a function of m, s1, x1, and x. Now use the method of undetermined coe cients tosolve for the values of a; b, c, and d that characterize the rational expectations equilibrium.[Hint: Everything follows once you solve for the ratio c=b. To do so, note that the ratio ofcoe cients on (x1) and s1 in your expression for the price must equal c=b.]c) What is the expected return (E [v1 Rp1]) on asset 1? Is this greater or less than the

expected return on asset 2 ? How, if at all, does the expected return on asset 1 depend onthe fraction of informed traders, ? What is the intuition for your results?

d) On average, which type of agents own more of asset 1? Explain.

e) How might you apply this model to explain home bias in portfolio choice? Discusshow you might test the models explanation for home bias if you could measure domesticand foreign investorsportfolio composition at each point in time and realized returns overtime.

If you get stuck on this problem, consult Maureen OHaras presidential address to theAmerican Finance Association, Liquidity and Price Discovery, Journal of Finance 58,13351354, 2003.

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3. Consider the following adaptation of Kyles (Econometrica 1986) static model withmarket orders. There is one risky asset with payo v that is normally distributed with zeromean and variance 2v. An insider, who has zero endowment of the risky asset, observesv and places a market order x. The insider has constant absolute risk aversion a, so shemaximizes

E[ exp(aW )];where W is terminal wealth.

Risk neutral market makers observe the total order ow x + u, where u is the demandof noise traders and is normally distributed with mean zero and variance 2u. Competitionbetween market makers is assumed to result in the market price

p = E[vjx+ u]:

The informed trader is assumed to behave strategically; that is, in deciding on her optimaltrade she takes the dependence of the price on the optimal order ow into account.

a) Assume that market price is a linear function of the total order ow,

p = (x+ u):

Solve for the informed traders demand as a function of v. What is the demand function inthe special case where the insider is risk neutral?

b) Show how is determined given the informed traders demand function. Show thata linear equilibrium exists, and state an equilibrium condition for in terms of exogenousparameters of the model. What is the equilibrium condition in the special case where theinsider is risk neutral?

c) Show that the equilibrium value of is decreasing in the insiders risk aversion a.Explain the economic intuition for this result. How would this result be aected if theinsider could observe the value of noise demand u before submitting her order?

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