Alberto Bisin

Lecture Notes on Financial Economics:

In�nite-Horizon Exchange Economies

September, 20091

1 Asset Pricing in In�nite Horizon models

Assume a representative-agent economy with one good. Let time be indexed byt = 0; 1; 2; :::: Uncertainty is captured by a probability space represented by atree. Suppose that there is no uncertainty at time 0 and call s0 the root of thetree. Without much loos of generality, we assume that each node has a constantnumber of successors, S. At generic node at time t is called st 2 St. Note thatthe dimensionality of St increases exponentially with time t (abusing notationit is in fact St).When a careful speci�cation of the underlying state space process is not

needed, we will revert to the usual notation in terms of stochastic processes.Let x := fxtg1t=0 denote a stochastic process for an agent�s consumption, wherext : S

t �! R+ is a random variable on the underlying probability space, for eacht. Similarly, let ! := f!tg1t=0 be a stochastic processes describing an agent�sendowments. Let 0 < � < 1 denote the discount factor.

1.1 Contingent Markets Economy

Suppose that at time zero, the agent can trade in contingent commodities. Letp := fptg1t=0 denote the stochastic process for prices, where pt : St �! R+, foreacht.

De�nition 1��x�i�i; p��is an Arrow-Debreu Equilibrium if

i. given p�;x�i 2 argmaxfu(x0) + E0[

P1t=1 �

tu(xt)]gs:t:P1

t=0 p�t (xt � !t) = 0

ii. andP

i x�i � !i = 0:

The notation does not make explicit that the agent chooses at time 0 awhole sequence of time and state contingent consumption allocations, that is,the whole sequence of x(st) for any st 2 St and any t � 0.

1Thanks to Francesc Ortega for research assistance.

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1.2 Financial Markets Economy

Suppose that throughout the uncertainty tree, there are J assets. We shall allowassets to be long-lived. In fact we shall assume they are and let the reader takecare of the straightforward extension in which some of the assets pay o¤ only ina �nite set of future times. Let z := fztg1t=1 denote the sequence of portfoliosof the representative agent, where zt : St �! RJ . Assets�payo¤s are capturedat each time t by the S�J matrix At. Furthermore, capital gains are qt� qt�1,and returns are Rt =

At+qtqt�1

.In a �nancial market economy agents do not trade at time 0 only. They in

fact, at each node st receive endowments and payo¤s from the portfolios theycarry from the previous node, they re-balance their portfolios and choose statecontingent consumption allocations for any of the successor nodes of st, whichwe denote st+1 j st.

De�nition 2 f�x�i; z�i

�i; q�g is a Financial Markets Equilibrium if

i. given q�; at each time t � 0�x�i; z�i

�2 argmaxfu(xt) + Et[

P1�=1 �

ju(xt+� jst)]gs:t:xt+� + q

�t+�zt+� = !t+� +At+�zt+��1;

for � = 0; 1; 2; :::; with z�1 = 0some no-Ponzi scheme condition

De�nition 3 ii.P

i x�i � !i = 0 and

Pi z�i = 0:

1.3 Conditional Asset Pricing

From the FOC of the agent�s problem, we obtain

qt = Et

��u0(xt+1)

u0(xt)At+1

�= Et (mt+1At+1) (1)

or,

1 = Et

��u0(xt+1)

u0(xt)Rt+1

�: (2)

Example 1. Consider a stock. Its payo¤ at any node can be seen as thedividend plus the capital gain, that is,

Rt+1 =qt+1 + dt+1

qt;

for some exogenously given dividend stream d. By plugging this payo¤ intoequation (1), we obtain the price of the stock at t.Example 2. For a call option on the stock, with strike price k at some

future period T > t, we can de�ne

At = 0; t < T; and AT = maxfqT � k; 0g

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De�ne now

mt;T =�T�tu0(xT )

u0(xt)

and observe that the price of the option is given by

qt = Et (mt;T maxfqT � k; 0g) ;

Note how the conditioning information drives the price of the option: the pricechanges with time, as information is revealed by approaching the execution pe-riod T .

Example 3. The risk-free rate is know at time t and therefore, equation(2) applied to a 1-period bond yields

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Rft+1= Et

��u0(xt+1)

u0(xt)

�:

Once again, note that the formula involves the conditional expectation at timet. Therefore, while the return of a risk free 1-period bond paying at t + 1 isknown at time t, the return of a risk free 1-period bond paying at t + 2 is notknown at time t. [.... relationship between 1 and � period bonds.... from thered Sargent book]

Conditional versions of the beta representation hold in this economy:

Et(Rt+1)�Rft+1 = �Covt(mt+1; Rt+1)

Et(mt+1)= (3)

=Covt(mt+1; Rt+1)

V art(mt+1)

��V art(mt+1)

Et(mt+1)

�=: �t�t:

1.3.1 Unconditional moment restrictions

Recall that our basic pricing equation is a conditional expectation:

qt = Et(mt+1At+1); (4)

In empirical work, it is convenient to test for unconditional moment restrictions.However, taking unconditional expectations of the previous equation implies inprinciple a much weaker statement about asset prices than equation (4):

E(qt) = E(mt+1At+1); (5)

where we have invoked the law of iterated expectations. It should be clear thatequation (4) implies but it is not implied by (5).The theorem in this section will tell us that actually there is a theoretical

way to test for our conditional moment condition by making a series of tests ofunconditional moment conditions.

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De�ne a stochastic process fitg1t=0 to be conformable if for each t, it belongsto the time-t information set of the agent. It then follows that for any suchprocess, we can write

itqt = Et(mt+1itAt+1)

and, by taking unconditional expectations,

E(itqt) = E(mt+1itAt+1):

This fact is important because for each conformable process, we obtain an ad-ditional testable implication that only involves unconditional moments. Ob-viously, all these implications are necessary conditions for our basic pricingequation to hold. The following result states that if we could test these uncon-ditional restrictions for all possible conformable processes then it would also besu¢ cient. We state it without proof.

Theorem 4 If E(xt+1it) = 0 for all it conformable then Et(xt+1) = 0:

By de�ning xt+1 = mt+1At+1 � qt; the theorem yields the desired result.

1.4 Predictability or returns

Recall the asset-pricing equation for stocks:

qt = Et (mt+1(qt+1 + dt+1)) :

It is sometimes argued that returns are predictable unless stock prices to followa random walk. (Where in turn predictability is interpreted as a property ofe¢ cient market hypothesis, a fancy name for the asset pricing theory exposed inthese notes). Is it so? No, unless strong extra assumptions are imposed Assumethat no dividends are paid and agents are risk neutral; then, for values of �close to 1 (realistic for short time periods), we have

qt = Et(qt+1):

That is, the stochastic process for stock prices is in fact a martingale. Next, forany f"tg such that Et("t+1) = 0 at all t, we can rewrite the previous equationas

qt+1 = qt+1 + "t+1:

This process is a random walk when vart("t+1) = � is constant over time.A more important observation is the fact that marginal utilities times asset

prices (a risk adjusted measure of asset prices) follow approximately a martingale(a weaker notion of lack of predictability). Again under no dividends,

u0(ct)qt = �Et(u0(ct+1)(qt+1 + dt+1));

which is a supermartingale and approximately a martingale for � close to 1.

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1.5 Fundamentals-driven asset prices

For assets whose payo¤ is made of a dividend and a capital gain, FOC dictate

qt = Et (mt+1(qt+1 + dt+1)) ;

where

mt+1 =�u0(ct+1)

u0(ct):

By iterating forward and making use of the Law of Iterated Expectations,

qt = limT�!1

Et

0@ TXj=1

mt;t+jdt+j

1A+ limT�!1

Et

0@ TXj=1

mt;t+jqt+j

1A ;As we shall see, in�nite horizon models (with in�nitely lived agents) usually

satisfy the no-bubbles condition, or

limT�!1

Et

0@ TXj=1

mt;t+jqt+j

1A = 0:

In that case, we say that asset prices are fully pinned down by fundamentalssince

qt = Et

0@ 1Xj=1

mt;t+jdt+j

1A :1.5.1 Conditional factor models and the conditional CAPM

[...from Cochrane...]

1.5.2 Frictions

He-ModestLuttmer

2 Bubbles: Santos and Woodford, Ecta 1997

Let N = X1t=0S

t be the set of nodes of the tree. Recall we denoted with s0

denote the root of the tree and with st an arbitrary node of the tree at time t.Denote by st � 1 the single (immediate) predecessor node to st. Use st+� jst toindicate that st+� is some successor of st, for � > 0:At each node, there are J securities traded.It is important that the notation includes Overlapping Generation economies.

We need therefore to account for �nitely lived agents. Let I(st) be the set ofagents which are active at node st. Let N i be the subset of nodes of the tree

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at which agent i is allowed to trade. Also, denote by Nithe terminal nodes for

agent i.The following assumptions will not be relaxed.

1. If an agent i is alive at some non-terminal node st, she is also alive atall the immediate successor nodes. That is, st�N inN i

=) fst+1�N :st+1jstg � N i:

2. The economy is connected across time and states: at any state there issome agent alive and non-terminal. Formally,

8st;9i : st 2 N inN i:

Assets are long lived. Let q : N �! RJ be the mapping de�ning the vectorof security prices at each node st. Similarly, let d : N �! RJ denote the vector-valued mapping that de�nes the dividends (in units of numeraire) that are paidby the assets at node st. We assume that d(st) � 0 for any st:Each of the households alive at s0 enters the markets with an initial endow-

ment of securities zi! � 0: Therefore, the initial net supply of assets is givenby

z! =X

i2I(s0)

zi!.

As assets are long lived, a supply z! of assets is available at any st:At each node st, each households in I(st) has an endowment of numeraire

good of !i(st) � 0. We shall assume that the economy has a well-de�nedaggregate endowment

!(st) =X

i2I(st)

!i(st) � 0

at each node st. This is the case, e.g., if I(st) is �nite, for any st: Taking intoaccount the dividends paid by securities in units of good, the aggregate goodsupply in the economy is given by

e!(st) = !(st) + d(st)z! � 0:The utility function of any agent i is written

U(x) =1Xt=0

�tXst

probst ui(x(st)):

De�ne the 1-period payo¤ vector (in units of numeraire) at node st by

A(st) = d(st) + q(st):

Agent i chooses, at each node st 2 N i a level of consumption xi(st) and a Jvector of securities zi(st) to hold at the end of trading, subject to the budgetconstraints:

xi(s0) + q(s0)zi(s0) � !i(s0) + q(s0)zi!;

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and at each node st 6= s0,

xi(st) + q(st)zi(st) � !i(st) +A(st)zi(st � 1);

with

xi(st) � 0

q(st)zi(st) � �Bi(st);

where Bi : N �! R+ indicates an exogenous and non-negative household spe-ci�c borrowing limit at each node. We assume households take the borrowinglimits as given, just as they take security prices as given.At equilibrium, markets clear: that is, at each st;X

i2I(st)

xi(st) = e!(st)X

i2I(st)

zi(st) = z!

Given the price process q, we say that no arbitrage opportunities exist atst if there is no z 2 RJ such that

A(st+1)z � 0; for all st+1jst;q(st)z � 0;

with at least one strict inequality.

Lemma 5 When q satis�es the no-arbitrage condition at st; there exists a setof state prices f�(st+1)g with �(st+1) > 0 for all st+1jst, such that the vectorof asset prices at st can be written as

q(st) =X

st+1jst�(st+1)A(st+1): (6)

Proof. As usual, proof follows from applying an appropriate separationtheorem.Applying the Lemma at any st we can construct a stochstic process � : N !

RS++: Let �(st) denote the set of such processes for the subtree with root st.

Only under complete markets is the set �(st) a singleton.As a remark, note that completeness is an endogenous property since one-

period payo¤s A contain asset prices. Therefore, the rank property which de�nescompleteness can only be assessed at each given equilibrium.

De�nition 6 For any state-price process � 2 �(st); de�ne the J vector offundamental values for the securities traded at node st by

f(st; �) =1X

T=t+1

XsT jst

�(sT )d(sT jst): (7)

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Observe that the fundamental value of a security is de�ned with referenceto a particular state-price process, however the following properties it displaysare true regardless of the state prices chosen.

Proposition 7 At each st 2 N , f(st; �) is well-de�ned for any � 2 �(st) andsatis�es

0 � f(st; �) � q(st):

Proof. First of all, 0 � f(st; �) follows directly from non-negativity of �,the dividend, and the price process. We therefore turn to f(st; �) � q(st): Fromequation (6), we have

q(st) =X

st+1jst�(st+1)d(st+1jst) +

Xst+1jst

�(st+1)q(st+1)

and, iterating on this equation we obtain

q(st) =

bTXT=t+1

XsT jst

�(sT )d(sT jst) +Xs bT jst

m(sbT )q(sbT )

for any bT > t: Since by construction, q(sbT ) is non-negative and � 2 �(st) isa positive state-price vector, the second term on the right-hand-side is non-negative. So,

q(st) �bTX

T=t+1

XsT jst

�(sT )d(sT jst); for any bT > t:and

q(st) �1X

T=t+1

XsT jst

�(sT )d(sT jst) = �(st)f(st; �)

We can correspondingly de�ne the vector of asset pricing bubbles as

�(st; �) = q(st)� f(st; �); (8)

for any � 2 �(st) for the J securities. It follows from the proposition that

0 � �(st; �) � q(st);

for any � 2 �(st): This corollary is known as the impossibility of negative bubblesresult. Substituting (8) and (7) into (6) yields

�(st) =X

st+1jst�(st+1)�(st+1):

This is known as the martingale property of bubbles: if there exists a (nonzero)price bubble on any security at date t, there must exist a bubble as well on the

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security at date T , with positive probability, at every date T > t. Furthermore,if there exists a bubble on any security at node st, then there must have existeda bubble as well on some security at every predecessor of the node st.What does this imply for securities with �nite maturity? Your answer must

depend on how you de�ne securities with �nite maturity in this context.In an economy with incomplete markets, the fundamental value need not be

the same for all state-price processes consistent with no arbitrage. But even inthis case, we can de�ne the range of variation in the fundamental value, giventhe restrictions imposed by no-arbitrage.Let x : N �! R+ denote a stream of non-negative payo¤. For any st, pick

any � 2 �(st) and de�ne the present value at st of x with respect to � by

Vx(st; �) =

1XT=t+1

XsT jst

�(sT )x(sT ).

Since this present value depends on the stochastic discount factor �, let usnow de�ne the bounds for the present value at st of dividends x.

De�nition 8 For any st, de�ne

�x(st) = inf

�2�(st)fVx(st; �)g

�x(st) = sup

�2�(st)fVx(st; �)g:

A few remarks follow from these de�nitions. First note that these de�nitionsare conditional on a given price process q since the set of no-arbitragestochastic discount factors are de�ned with respect to q. Next observethat, for any security with payo¤ process xj,

�xj (st) � f j(st; �) � �xj (st), for all � 2 �(st), and

�xj (st) < qj implies that there is a pricing bubble for the security with payo¤process xj .

Recall that to rule out Ponzi schemes when agents are in�nitely lived, alower bound on individual wealth is needed. Let us de�ne a particular type ofborrowing limit.

De�nition 9 An agent�s borrowing ability is only limited by her ability to repayout of her own future endowment if

Bi(st) = �~!i(st); (9)

for each st�N inN i.

It can be shown that these borrowing limits never bind at any �nite date(see Magill-Quinzii, Econometrica, 94), but rather only constrain the asymptotic

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behavior of a household�s debt.2 An important consequence of this speci�cationis the following.

Proposition 10 Suppose that agent i has borrowing limits of the form (9).Then the existence of a solution to the agent�s problem for given prices q impliesthat �~!i(s

t) < 1; at each st 2 N i; so that the borrowing limit is �nite at eachnode.

This is because, if agent i can borrow o¤ of the value of ~!i; this value mustbe �nite at equilibrium prices for the agent�s problem to be well-de�ned. Thismust be the case, in fact, for the equilibrium price of ~!i (recall ~!i is traded)and hence it must be that �(st) < 1 (recall that for any traded process xj ;�xj (s

t) � �xj (st) � qj): We can now prove the following fundamental lemma.

Lemma 11 Consider an equilibrium fx�i; z�i; qg. Suppose that the (supremumof the) value of aggregate wealth is �nite, i.e., �~!(st) < 1. Suppose alsothat there exists a bubble on some security in positive net supply at st so that�(st)z(st) > 0. Then, 8K > 0; there exists a time T and sT jst such that

�(sT )z(sT ) > KXsT jst

�(sT )e!(sT ):Proof. The martingale property of pricing bubbles implies,

�(st)z(st) =XsT jst

m(sT )�(sT )z(sT )

and hence �(st)z(st) explodes with positive probability. On the other hand,PsT jst �(s

T )e!(sT ) must converge to 0 in T !1 to guarantee that �~!(st) <1:

That is, there is a positive probability that the total size of the bubble onthe securities becomes an arbitrarily large multiple of the value of the aggregatesupply of goods in the economy. The proof exploits crucially the martingaleproperty of bubbles. It follows from this result that some agent must accumulatevast wealth.We already learned that no bubbles can arise in securities with �nite matu-

rity. The next theorem shows that no bubbles can arise to securities in positivenet supply as long as we are at equilibria with �nite aggregate wealth. Theproof uses the nonoptimality of the behavior implied by the previous lemma.

Theorem 12 Consider an equilibrium fx�i; z�i; qg. Suppose that at each nodest 2 N; there exists � 2 �(st) such that V~!(st; �) <1: Then

qj(sT ) = f j(sT ; �);

for all sT jst and � 2 �(st), for each security j traded at sT that has positivenet supply, zj! > 0.

2They are equivalent to requiring that the consumption process lies in the space of mea-surable bounded sequences. In the case of �nitely lived agents, these borrowing limits areequivalent to imposing no-borrowing at all nonterminal nodes.

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This is a crucial: if an agent�s endowment can be traded, then its value ison the right hand side of the present value budget constraint of the agent, andhence it must be �nite. Note that if we have that at equilibrium �~!(s

t) < 1;the condition of the theorem is satis�ed.The next two corollaries to the theorem provide conditions on the primitives

of the model that guarantee that the value of aggregate wealth is �nite at anyequilibrium.

Corollary 13 Suppose that there exists a portfolio bz�RJ+ such thatd(stjs0)bz � ~!(st); 8st 2 N:

Then the theorem holds at any equilibrium.

Intuitively, if the existing securities allow such a portfolio bz to be formed,it must have a �nite price at any equilibrium. But since the dividends paidby this portfolio are higher at every state than the aggregate endowment, theequilibrium value of the aggregate endowment is bounded by a �nite number.

Corollary 14 Suppose that there exists an (in�nitely lived) agent i and an" > 0 such that i) !i(st) � "!(st); 8st 2 N and ii) Bi(st) = �!i(s

t); 8st 2 Nand for all i. Then the theorem holds at any equilibrium.

Again, the result follows because in equilibrium, "!(st) must have a �nitevalue as it appears on the right hand side of agent i�s budget set. If a positivefraction of aggregate wealth has a �nite value in equilibrium, then aggregatewealth has �nite value.As a remark, note that these two corollaries share the same spirit and some-

what imply that bubbles are not a robust equilibrium phenomenon (for securitiesin positive net supply).

2.1 (Famous) Theoretical Examples of Bubbles

Recall that �at money is a security that pays no dividends. Its only returncomes from paying one unit of itself in the next period. Therefore if �at moneyis in net supply and has a positive price in equilibrium, that is a bubble. Thefollowing two models have equilibria with such a property.

2.1.1 Samuelson (1958)�s OLG model

Consider an economy in which[st2N

I(st) is (countably) in�nite, even though

I(st) is �nite for any st 2 N: In this case even if �~!i(st) <1, it is still possiblethat �~!(s

t) = 1 (and hence that �~!(st) = 1). The theorem does not applyand bubbles are possible.

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2.1.2 Bewley (1980)�s turnpike model

Consider the case in which stringent borrowing limits are imposed on trading,i.e., Bi(st) < �~!i(s

t). In this case, nothing will exclude the possibility that�~!i(s

t) =1. The theorem does not apply and bubbles are possible.

2.1.3 More recent examples

Here�s a list.

Boyan Jovanovic (2007), Bubbles in Prices of Exhaustible Resources, NBERWorking Paper No. 13320, http://www.nber.org/papers/w13320

Harrison Hong, Jose Scheinkman, Wei Xiong (2006), Asset Float and Spec-ulative Bubbles, Journal of Finance, American Finance Association, vol.61(3), pages 1073-1117. http://www.princeton.edu/~wxiong/papers/�oat.pdf

Dilip Abreu and Markus Brunnermeier (2003), Bubbles and Crashes, Econo-metrica, 71(1), 173-204, http://www.princeton.edu/~markus/research/papers/bubbles_crashes.htm

Franklin Allen, Stephen Morris, and Hyun Song Shin (2003), Beauty Contests,Bubbles and Iterated Expectations in Asset Markets, Cowles FoundationDiscussion Paper 1406, http://cowles.econ.yale.edu/P/cd/d14a/d1406.pdf

3 Idiosynchratic shocks economies

The class of economies we studied for bubbles has a �nite number of (types of)agents at each node of the tree N : I(st) is �nite for any st: In these economiesthe stochastic structure represented by the tree N refers to the whole economy:all agents i 2 I(st) at time t face the realization st: However, we often need toexplicit the composition of the types, that is, to explicit them notationally asan in�nity of ex-ante identical agents , so as to make (ab)use of the Law of largenumbers.In this section we show how to extend the notation to the case of idiosyn-

chratic shocks economies. For simplicity, but without loss of generality, weconsider the case in which all agents in the economy are of the same type: thatis, agents are ex-ante identical in terms of preferences and stochastic processfor endowments. Ex-post, however, the realization of their endowments are dif-ferent across agents. Essentially, we now think of the tree N as representingthe stochastic process faced by each individual agent. Let each node st have Ssuccessors, that is, st+1jst = f1; 2; :::; Sg : Since macroeconomics is typically setin terms of Markov processes, we adopt the assumption:

prob(st+1 = s0jst = s) = prob (s0js)

Obviously prob (sjs0) de�nes a transition probability of a Markov chain withstate space f1; 2; :::; Sg. We assume the Markov chain is recurrent (e.g., it is

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su¢ cient that prob (s0js) > 0, for any s0; s 2 f1; 2; :::; Sg) so that it has astationary distribution �(s):

�(s0) =Xs2S

prob (s0js)�(s)

The (ab)use of the Law of large number consists in assuming the following:

The fraction of agents whose realization of the individual shock is sat time t and s0 at t+ 1 is prob (s0js); for any t � 0:

In these economies, therefore, we can conveniently de�ne individual endow-ments as a map ! : f1; 2; :::; Sg �! R+, for each time t � 0. Similarly, assets�dividends are d : f1; 2; :::; Sg �! RJ+ ; a J-dimensional Markov process in ourprobability space. Prices are a vector sequence qt 2 RJ+; for any t � 0: Eachindividual agent problem is then more easily written recursively:

vt(z; s) = maxz0��B

u (d(s)z + !(s)� qtz0) + �Xs02S

prob (s0js)vt+1(z0; s0);

where �B is the natural borrowing limit, as de�ned, e.g., in the section onbubbles. The policy function is

zt+1 = gt(zt; st; qt);

andxt(zt; st) = d(st)zt + !(st)� qtgt(zt; st; qt)

Construct then the stationary distribution

�t+1(zt+1; st+1) =Xst2S

prob (st+1jst)Zzt:zt+1=gt(zt;st;qt)

�t(zt; st)dzt

Using the ab(use) of the Law of large numbers, therefore, goods and �nancialmarket clearing at each time t take the form:

Xst2S

�t(zt; st) (xt(zt; st)� !(st)) = 0

Xs2S

Z�t(zt; st)gt(zt; st; qt)dzt = 0;

respectively.Summarizing:

De�nition 15 A Financial markets equilibrium for the economy with idio-synchratic risk is represented by sequences of maps xt : S � RJ �! R+;

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gt : S�RJ �! RJ ; and a sequence of vectors qt 2 RJ+, for any t � 0, such thatzt+1 = gt(zt; st; qt) solves:

vt(z; s) = maxz0��B

u (d(s)z + !(s)� qtz0) + �Xs02S

prob (s0js)vt+1(z0; s0);

andxt(zt; st) = d(st)zt + !(st)� qtgt(zt; st; qt);

Furthermore, �nancial markets clear:3Xs2S

Z�t(zt; st)gt(zt; st; qt)dzt = 0:

Consider now an extension of the previous section�s economy, which allowsfor aggregate risk, that is, a stochastic process for shocks at 2 A = f1; :::; Agwhich a¤ect all the agents. Essentially, we keep having the tree N as the rep-resentation of the stochastic process faced by each individual agent. But nowthink of each node on the tree N as a couple at; st 2 A � S with have A � Ssuccessors, that is, at+1; st+1jat; st 2 A � S: The transition probability for theMarkov processes is:

prob(at+1 = a0; st+1 = s0jat = a; st = s) = prob (a0; s0ja; s)

with stationary distribution �(a; s): Each individual agent problem is then moreeasily written recursively:

vt(z; a; s) = maxz0��B

u (d(a; s)z + !(a; s)� qt(a)z0)+�X

a0;s02S�Aprob (a0; s0ja; s)vt+1(z0; a0; s0);

and the policy function is

zt+1 = gt(zt; at; st)

Summarizing:

De�nition 16 A Financial markets equilibrium for the economy with idiosyn-chratic and aggregate risk is represented by sequences of maps xt : A�S�RJ �!R+; gt : A � S � RJ �! RJ ; and qt : A �! RJ+, for any t � 0, such thatzt+1 = gt(zt; at; st) solves:

vt(z; a; s) = maxz0��B

u (d(a; s)z + !(a; s)� qt(a)z0)+�X

a0;s02S�Aprob (a0; s0ja; s)vt+1(z0; a0; s0);

3Goods market clearing,Xst2S

�t(zt; st) (xt(zt; st)� !(st)) = 0 ;

is redundant.

14

andxt(zt; at; st) = d(at; st)zt + !(at; st)� qt(at)gt(zt; at; st)

Furthermore, �nancial markets clear:Xa;s2S�A

Z�t(zt; at; st)gt(zt; at; st)dzt = 0:

Typically, these economies have well-de�ned stationary equilibria.

De�nition 17 A stationary Financial markets equilibrium for the economywith idiosynchratic and aggregate risk is represented by maps x : A�S�RJ �!R+; g : A� S �RJ �! RJ ; and q : A �! RJ+ such that z

0 = g(z; a; s) solves:

v(z; a; s) = maxz0��B

u (d(a; s)z + !(a; s)� q(a)z0)+�Xs02S

prob (a0; s0ja; s)v(z0; a0; s0);

andx(z; a; s) = d(a; s)z + !(a; s)� q(a)g(z; a; s)

Furthermore, �nancial markets clear:Xa;s2A�S

Zg(z; a; s)�(z; a; s)dz = 0;

for a stationary distribution �(z; a; s) which satis�es:

�(z0; a0; s0) =X

a;s2S�Aprob (a0; s0ja; s)

Zz:z0=g(z;a;s;q)

�(z; a; s)dz:

The following result is not out there in the literature (I think). But itshould be a consequence of the constrained e¢ ciency result for incomplete mar-ket economies with one good. Can you prove it?

Proposition 18 A stationary Financial market equilibrium for an incompletemarket economy with idiosynchratic and aggregate risk is constrained Paretoe¢ cient. On the other hand, a Financial market equilibrium for an incompletemarket economy with idiosynchratic and aggregate risk (along the transitionpath) is generically constrained Pareto ine¢ cient.

The constraint ine¢ ciency result for Bewley economies, due to Davila, Hong,Krusell, Rios-Rull (2005), applies to production economies; see later.

3.1 When do Incomplete Markets matter?

In a series of papers, Telmer (1993), Aiyagari (1994) and Krusell and Smith(1998) among others, di¤erent authors have found support for a puzzling result.Even though theoretically the completeness of �nancial markets a¤ects equilib-rium allocations and prices in a fundamental way (e.g., equilibria are typically

15

e¢ cient if and only if �nancial markets are complete), they do not seem tomatter signi�cantly in calibrations.In fact, we can show theoretically that, when agents are in�nitely lived and

their endowments are stationary and idiosynchratic, incomplete markets tend tomatter little (where "little" is precisely de�ned). This is the result from Levineand Zame, Econometrica, 2002. The intuition for this result is straightforward:over long horizons market incompleteness may not matter if traders can selfinsure i.e., if they can borrow in bad times and save in good times.Consider the economy with idiosynchratic (no aggregate) risk in the previous

section, extended to allow for a �nite set of types i 2 I (each of measure 1):Let !i and ! denote the long-run average endowment (permanent income) foragent i and for the aggregate economy, respectively:

!i =Xs2S

�(s)!i(s);

! =Xi2I

Xs2S

�(s)!i(s):

Assume that only a bond is traded in the economy ( J = 1), that is an asset withpayo¤ d(s) = 1; for any s:It is easy to show that, the Pareto e¢ cient allocationsof this economy are given by the I-tuples of �xed shares of the constant long-runaverage endowment ! =

Pi !

i. In particular, the complete market equilibriumallocation (hence Pareto e¢ cient) for this economy is characterized by eachagent i consuming his/her long-run average endowment (permanent income) ateach node in N:The next theorem shows that, for an appropriate de�nition of "closeness",

when markets are incomplete, equilibrium allocations are "close" to ones whereeach agent i consumes his/her long-run average endowment (permanent income)at each node. To this end, letN"

t � N denote the set�st 2 St

��xit(st)� !i > " for some ifor some equilibrium allocation xit(st): We say that the equilibrium stochasticprocess xi is " close" to !i; denoted xi � !i for any i if,

for any "; � > 0; there exists a � su¢ ciently close to 1 such that

� > (1� �)1Xt=0

�tXst2N"

t

prob(st��s0 )

In other words, we say that the allocation process xit is "close" to agent i�s long-run average endowment if the time-discounted probability that consumptiondeviates from the perfect risk sharing allocation by more than a given amountis small.

Theorem 19 Suppose Dui(x) is (weakly) convex, for any i 2 I: Any Financialmarket equilibrium allocation process of the economy with idiosynchratic risk,x�i; is "close" to perfect risk sharing,

x�i � !i.

16

The proof involves constructing a budget feasible plan, a stochastic processxit for any i; whose utility is almost that of constant average consumption:

lim�!1

�����E 1Xt=0

ui(xit)

!� 1

1� � u(!i)

����� = 0A crucial step in the argument is establishing that the riskless interest rate,1qtis bounded above, with a bound close to 1, if Dui(x) is (weakly) convex,

for any i 2 I (that is, if all agents have preferences for precautionary savings).This is important because the budget feasible plan constructed in the proof is�nanced by borrowing, and a low interest rate makes borrowing easy. A simplecontinuity argument then implies the result that x�i � !i:In general, in the presence of aggregate risk, market incompleteness matters

even if endowment processes are stationary (i.e., shocks are transitory). Thereason is the following. When there is aggregate risk, the upper bound onthe interest rate need not obtain; when the aggregate endowment is low, manytraders will want to borrow, and this demand for loans may drive up the risklessinterest rate. A high interest rate interferes with risk sharing because it makesborrowing di¢ cult. Summing up, aggregate risk matters because it a¤ects assetprices.When there is more than one consumption good, market incompleteness

matters again, even without aggregate risk. The reason is that commodityprices provide another source of untraded risk.We conclude that in a one-good economy populated by in�nitely-lived, pa-

tient agents, market incompleteness will not matter if shocks are transitory andrisk is purely idiosyncratic. When there is aggregate uncertainty or more thanone consumption good, market incompleteness matters, in general.It is clear that Levine and Zame�s argument requires stationary individ-

ual endowments processes (otherwise !i is not de�ned). Constantinedes andDu¢ e, Journal of Political Economy 1996, show a partial converse: a particu-lar non-stationary individual endowment process, when markets are incomplete,essentially any stochastic discount factor in terms of aggregate consumption.Consider the idiosynchratic and aggregate risk economy introduced above.

Assume agents have identical CRRA instantaneous preferences with risk aver-sion parameter �:

u(x) =1

1� �x1��

Assume agents can trade two assets, i) a long-lived equity in positive (= 1) netsupply, paying dividend dt = d(at) > 0; and ii) a bond paying 1 in every statea; s 2 A� S: Let xt denote aggregate consumption at time t: De�ne

!it = �itxt � dt; for any t

17

where

�it = exp

"tX

�=1

�i�y� �

(y� )2

2

!#

yt =

s2

�(�+ 1)

�log

mt

mt�1+ � + � log

xtxt�1

� 12

;

and �i� are normal i.i.d. shocks, mt is a stochastic discount factor consistentwith no-arbitrage. Note that the fact that the driving shocks �i� a¤ect cumula-tively the individual share of aggregate income, as opposed to individual income,implies that individual income is not stationary.It follows that: X

i

�it =

Z�(s)prob(s)ds = 1

MRSit =mt

mt�1; for any i

and hence at equilibrium each agent consumes his own endowment. Finally,note that

i) the stochastic discount factor mt+1

mtcan be expressed as a function of (the ra-

tion of) aggregate consumption xt+1xt

with a degree of freedom represented

by (yt+1)2:

E

Rt+1�

�xt+1xt

��aexp

��(�+ 1)

2(yt+1)

2

�!= 1;

for any return Rt+1 in the span of the asset space; and

ii) (yt+1)2 is the variance of the cross-sectional distribution of log

xit+1xt+1

xitxt

:

log

xit+1xt+1

xitxt

= log�it+1

�it� N

� (yt+1)

2

2; (yt+1)

2

!:

4 Macro with incomplete markets

In this section we consider two economies with incomplete markets and idiosyn-chratic shocks which have been studied in macroeconomics. The �rst, referredto as Bewley economies, are economies characterized by:

agents face stationary idiosynchratic endowment shocks (with/out an aggregatecomponent), but

18

can only trade a riskless bond, typically with a no-short-sales constraint.

The second class of economies, referred to as individual risk economies, arecharacterized by:

agents face stationary idiosynchratic shocks (with/out an aggregate compo-nent) on the rate of return of savings, but

can only trade a riskless bond, typically with a no-short-sales constraint.

Both Bewley economies and individual risk economies can be somewhatextended to allow for production. We discuss these extension in a future chapter.

4.1 Bewley economies

The prototypical Bewley economy is an economy with idiosynchratic shocks inwhich asset trading is restricted to a bond, which trades at time t for a price qtnormalized 1 for any t � 0; and pays pays 1+ rt+1 at time t+1: This economy,originally studied by Huggett (1993); see Ljungvist-Sargent (2004), ch. 17.This economy is straightforwardly modi�ed to the case in which agents face

a no-borrowing constraint, zt � 0; and the rate of return on savings is anexogenous sequence rt; see Aiyagari (1994) and Ljungvist and Sargent (2004),ch. 17. An equilibrium is still characterized by a policy function of the form

zt+1 = gt(zt; st);4

and a distribution �t(zt; st); de�ned recursively from an initial given distribution�0(z0; s0): The distribution of wealth at time t is:

�t(zt) =Xst2S

�t(zt; st):

The limit distribution of wealth, if it exists, satis�es:

�(z) = limt!1

�t(zt):

Proposition 20 The limit distribution of wealth in Bewley economies, �(z);has thin tails, that is, all its moments are well de�ned.

This is perhaps problematic in lieau of the evidence that the distribution ofwealth in the U.S. (as well as in many other developed countries; see Benhabib-Bisin-Zhu (2009) for a summary of this evidence).

4We drop the dependence of the policy function from the whole sequence rt; for notationalsimplicity.

19

4.2 Bewley economies with production

A straightforward modi�cation/reinterpretation/extension of this economy, hasproduction to endogeneize the rate of return (on capital; re-interpret wealth ascapital) and a wage rate. Let F (Kt; Nt) the aggregate production function attime t, in terms of aggregate capital Kt and labor Nt: Assume e.g., that laborsupply is �xed and !t(s) = wts, where wt is the wage rate at time t, and theshock s is interpreted as labor productivity. Then, de�ning

Kt+1 =

Z�t+1(zt+1)dzt+1; Nt = N =

Xs2S

�(s)s

we have

rt =@F (Kt; N)

@Kt; wt =

@F (Kt; N)

@N:

Aggregate shocks to the production function can be easily added. Typically wewrite it as

atF (Kt; Nt):

Proposition 21 A Financial market equilibrium for a Bewley economy withproduction is generically constrained Pareto ine¢ cient.

This is the result, mentioned above, and due to Davila, Hong, Krusell, Rios-Rull (2005). The proof is obtained essentially by comparing the �rst orderconditions at equilibrium with those at a constrained Pareto optimum. We onlyneed to prove the result at an (ergodic) stationary equilibrium. Let �(z) denotethe stationary distribution of wealth at equilibrium. Let the stationary interestrate and wage rate be denoted, respectively, r(K) and w(K) to make theirdependence from the aggregate wealth of the economy explicit:5 r = @F (K;N)

@K

and w = @F (K;N)@N .

The �rst order conditions at equilibrium include

d

dxu (r(K)z + w(K)s� z0) � �r(K)

Xs02S

prob (s0js) ddxu (r(K)z0 + w(K)s0 � g(z; s)) ;

while the �rst order conditions at a Pareto e¢ cient allocation include

d

dxu (r(K)z + w(K)s� z0) � �r(K)

Xs02S

prob (s0js) ddxu (r(K)z0 + w(K)s0 � g(z; s))+

+�Xs02S

prob (s0js) ddxu (r(K)z0 + w(K)s0 � g(z; s))

�dr(K)

dKz0 + s0

dw(K)

dK

��(z)

5Formally, and in general, they depend on the stationary distribution �(z; s). But in thiseconomy, where the production function aggregates all individual wealth, this reduces to adependence on K =

R�(z)dz:

20

As we are used to do by now, we avoid the details of the argument showingthat, generically, the �rst order conditions at equilibrium and at the Paretoe¢ cient allocation are di¤erent; that is, that the extra term in the conditionsfor Pareto e¢ ciency

+�Xs02S

prob (s0js) ddxu (r(K)z0 + w(K)s0 � g(z; s))

�dr(K)

dKz0 + s0

dw(K)

dK

��(z)

induces generically a di¤erent policy function.Intuitively, constrained Pareto ine¢ ciency follows from r and w being en-

dogenous.As for the limit distribution of wealth, the same result holds for Bewley

economies with production as with standard Bewley economies, as the interestrate is constant (or only dependent on aggregate risk at):

Proposition 22 The limit distribution of wealth in Bewley economies with pro-duction, �(z); has thin tails, that is, all its moments are well de�ned.

4.3 Investment risk economies

Suppose instead each agent faces an idiosynchratic exogenous rate of return onsavings, a mapping r : f1; 2; :::; Sg �! R+. We maintain the assumption thateach agent can only save (not borrow) at the rate r: zt � 0 for any t � 0. Giventhe process r;6 each agent solves:

vt(z; s) = maxz0�0

u ((1 + r(s)) z + !(s)� z0) + �Xs02s

prob (s0js)vt(z0; s0)

The solution of this problem is a policy function of the form:

zt+1 = g(zt; st):

Let

�t+1(zt+1; st+1) =Xst2S

prob (s0js)Zzt:zt+1=g(z;s)

(zt; st)�t(zt; st)dzt; for any t � 0:

Thenn the distribution of wealth at time t+1 in this economy is de�ned recur-sively, from an initial given distribution �0(z0; s0):

�t+1(zt+1) =X

st+12S�t+1(zt+1; st+1)

The limit distribution of wealth, if it exists, satis�es:

�(z) = limt!1

�t(z):

6Formally, the realization for s�1 is also given. Assume also agents are endowed with noportfolio positions on bonds at time 0: zi(s�1) = 0; for all i 2 I:

21

Conjecture 23 The limit distribution of wealth in Investment risk economies,�(z); is a power law in the tail:

limz!1

�(z) / z��; � > 1:

As a consequence it displays thick tails (let k be the smallest integer such that� < k, then the Pareto distribution with tail z�� has no moments of order kand higher).

4.4 Other incomplete market economies

Kubler-SchmeddersHeaton-Lucas

22