Financial Economics - New York University notes sept 2010.pdf · Besides providing an introduction to Financial economics, these notes havethereforealsotheambitionof suggestingauseful

Financial EconomicsLecture notes

Alberto Bisin

Dept. of Economics

NYU

September 25, 2010

Contents

Preface ix

1 Introduction 1

2 Two-period economies 32.1 Arrow-Debreu economies . . . . . . . . . . . . . . . . . . . . . 32.2 Financial market economies . . . . . . . . . . . . . . . . . . . 6

2.2.1 The stochastic discount factor . . . . . . . . . . . . . . 92.2.2 Arrow theorem . . . . . . . . . . . . . . . . . . . . . . 112.2.3 Aggregation . . . . . . . . . . . . . . . . . . . . . . . . 122.2.4 Asset pricing . . . . . . . . . . . . . . . . . . . . . . . 172.2.5 Pareto optimality . . . . . . . . . . . . . . . . . . . . . 22

2.3 Corporate �nance economies . . . . . . . . . . . . . . . . . . . 252.4 Asymmetric information economies . . . . . . . . . . . . . . . 34

2.4.1 Moral hazard insurance economies . . . . . . . . . . . . 352.4.2 Corporate agency economies . . . . . . . . . . . . . . . 39

3 In�nite-horizon economies 413.1 Arrow-Debreu economies . . . . . . . . . . . . . . . . . . . . . 413.2 Financial markets economies . . . . . . . . . . . . . . . . . . . 42

3.2.1 Asset pricing . . . . . . . . . . . . . . . . . . . . . . . 433.2.2 Bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . 463.2.3 Pareto optimality . . . . . . . . . . . . . . . . . . . . . 55

3.3 Bewley economies . . . . . . . . . . . . . . . . . . . . . . . . . 623.3.1 Earning risk . . . . . . . . . . . . . . . . . . . . . . . . 633.3.2 Investment risk . . . . . . . . . . . . . . . . . . . . . . 643.3.3 Limit incomplete market economies . . . . . . . . . . . 65

3.4 Asymmetric information economies . . . . . . . . . . . . . . . 67

v

vi CONTENTS

3.4.1 Lack of Commitment Economies . . . . . . . . . . . . . 68

Preface

These notes constitute the material for the graduate course on FinancialEconomics I at NYU.

ix

Chapter 1

Introduction

The aim of these lecture notes is to provide an introduction to Asset pricingand Corporate �nance in the context of general equilibrium models. Whilenot standard, this approach is consistent and related to the practice of macro-economics and it has the advantage of facilitating a coherent understandingof �nance in both its asset pricing and of corporate �nance manifestations.Besides providing an introduction to Financial economics, these notes

have therefore also the ambition of suggesting a useful approach to theoreticaland empirical research in the �eld.We will �rst (very quickly) review two-period economies, an environment

in which concepts can be de�ned and results proved with minimal notation.We will then study in�nite horizon economies, the typical workhorse in �-nance and macroeconomics. In the context of in�nite horizon economies wewill study exchange economies without commitment and with asymmetricinformation: these economies are rapidly becoming the frontier in macroeco-nomics. Finally, we shall also study production economies without commit-ment and with asymmetric information. While not much studied yet, theseare the natural environments to integrate asset pricing and corporate �nance.

1

Chapter 2

Two-period economies

In a two-period pure exchange economy we study �nancial market equilib-ria. In particular, we study the welfare properties of equilibria and theirimplications in terms of asset pricing.In this context, as a foundation for macroeconomics and �nancial eco-

nomics, we study su¢ cient conditions for aggregation, so that the standardanalysis of one-good economies is without loss of generality, su¢ cient condi-tions for the representative agent theorem, so that the standard analysis ofsingle agent economies is without loss of generality.The No-arbitrage theorem and the Arrow theorem on the decentraliza-

tion of equilibria of state and time contingent good economies via �nancialmarkets are introduced as useful means to characterize �nancial market equi-libria.As this chapter is intended as a review of known material, we shall not

provide any proof, for which we shall refer the reader to Bisin (2010).

2.1 Arrow-Debreu economies

Consider an economy extending for 2 periods, t = 0; 1. Let i 2 f1; :::; Igdenote agents and l 2 f1; :::; Lg physical goods of the economy. In addition,the state of the world at time t = 1 is uncertain. Let f1; :::; Sg denote thestate space of the economy at t = 1. For notational convenience we typicallyidentify t = 0 with s = 0, so that the index s runs from 0 to S:De�ne n = L(S+1): The consumption space is denoted then by X � Rn+.

Each agent is endowed with a vector !i = (!i0; !i1; :::; !

iS), where !

is 2 RL+;

3

4 CHAPTER 2 TWO-PERIOD ECONOMIES

for any s = 0; :::S. Let ui : X �! R denote agent i�s utility function. Wewill assume:

Assumption 1 !i 2 Rn++ for all i

Assumption 2 ui is continuous, strongly monotonic, strictly quasiconcaveand smooth, for all i (see Magill-Quinzii, p.50 for de�nitions and details).Furthermore, ui has a Von Neumann-Morgernstern representation:

ui(xi) = ui(xi0) +SXs=1

probsui(xis)

Suppose now that at time 0, agents can buy contingent commodities.That is, contracts for the delivery of goods at time 1 contingently to therealization of uncertainty. Denote by xi = (xi0; x

i1; :::; x

iS) the vector of all

such contingent commodities purchased by agent i at time 0, where xis 2 RL+;for any s = 0; :::; S: Also, let x = (x1; :::; xI):Let � = (�0; �1; :::; �S); where �s 2 RL+ for each s; denote the price of

state contingent commodities; that is, for a price �ls agents trade at time 0the delivery in state s of one unit of good l:Under the assumption that the markets for all contingent commodities

are open at time 0, agent i�s budget constraint can be written as1

�0(xi0 � !i0) +

SXs=0

�s(xis � !is) = 0 (2.1)

De�nition 2.1.1 An Arrow-Debreu equilibrium is a (x�; ��) such that

1: x�i 2 argmaxui(xi) s.t. ��0(xi0 � !i0) +

SXs=0

��s(xis � !is) = 0; and

2:

IXi=1

x�is � !is = 0, for any s = 0; 1; :::; S

1We write the budget constraint with equality. This is without loss of generality undermonotonicity of preferences, an assumption we shall maintain.

2.1 ARROW-DEBREU ECONOMIES 5

Observe that the dynamic and uncertain nature of the economy (con-sumption occurs at di¤erent times t = 0; 1 and states s 2 S) does notmanifests itself in the analysis: a consumption good l at a time t and state sis treated simply as a di¤erent commodity than the same consumption goodl at a di¤erent time t0 or at the same time t but di¤erent state s0. This isthe simple trick introduced in Debreu�s last chapter of the Theory of Value.It has the fundamental implication that the standard theory and results ofstatic equilibrium economies can be applied without change to our dynamic)environment. In particular, then, under the standard set of assumptions onpreferences and endowments, an equilibrium exists and the First and SecondWelfare Theorems hold.2

Theorem 2.1.2 Any Arrow-Debreu equilibrium allocation x� is Pareto Op-timal.

Recall that the proof exploits strict monotonicity of preferences and littleelse.Arrow-Debreu economies are easily extended to account for production.

Suppose h = 1; ::::H (types of) �rms produce at date 1 using as only inputthe amount kh of the numeraire good invested as capital at time 0: Theoutput depends on kh � 0 according to the function yhs = fh(kh; s) 2 RL+;for any h = 1; ::::H. We assume that fh(kh; s) is continuously di¤erentiable,increasing and concave in kh; and it satis�es fh(0; s) = 0; for any h = 1; ::::Hand any s = 1; :::; S:

De�nition 2.1.3 An Arrow-Debreu equilibrium of the economy with produc-tion is a (x�; k�; y�; ��) such that

1: x�i 2 argmaxui(xi) s.t. ��0(xi0 � !i0) +

SXs=0

��s(xis � !is) = 0;

2: (k�h; y�h) 2 argmax��0kh +SXs=0

��sy�h, s.t. yhs = f

h(kh; s); for any s = 1; :::; S

3:

IXi=1

x�i0 � !i0 = �HXh=1

k�hs ; andIXi=1

x�is � !is =HXh=1

y�hs , for any s = 1; :::; S

The First and Second Welfare Theorems are straightforwardly extended.2Having set de�nitions for 2-periods Arrow-Debreu economies, it should be apparent

how a generalization to any �nite T -periods economies is in fact e¤ectively straightforward.In�nite horizon will be dealt with in successive notes.


2.2 Financial market economies

Consider the 2-period economy just introduced. Suppose now contingentcommodities are not traded. Instead, agents can trade in spot markets andin j 2 f1; :::; Jg assets. An asset j is a promise to pay ajs � 0 units of goodl = 1 in state s = 1; :::; S.3 Let aj = (a

j1; :::; a

jS): To summarize the payo¤s

of all the available assets, de�ne the S � J asset payo¤ matrix

A =

0BBB@a11 ::: a

J1

::: :::

a1S ::: aJS

1CCCA :It will be convenient to de�ne as to be the s-th row of the matrix. Note thatit contains the payo¤ of each of the assets in state s.Let p = (p0; p1; :::; pS), where ps 2 RL+ for each s, denote the spot price

vector for goods. That is, for a price pls agents trade one unit of good lin state s: Recall the de�nition of prices for state contingent commodities inArrow-Debreu economies, denoted �: Note the di¤erence. Let good l = 1at each date and state represent the numeraire; that is, p1s = 1, for alls = 0; :::; S.Let xisl denote the amount of good l that agent i consumes in good s. Let

q = (q1; :::; qJ) 2 RJ+, denote the prices for the assets.4 Note that the pricesof assets are non-negative, as we normalized asset payo¤ to be non-negative.Given prices (p; q) and the asset structure A, any agent i picks a con-

sumption vector xi 2 X and a portfolio zi 2 RJ to

maxui(xi)

s.t.

p0(xi0 � !i0) = �qzi

ps(xis � !is) = Asz

i; for s = 1; :::S:

3The non-negativity restriction on asset payo¤s is just for notational simplicity.4Quantities will be row vectors and prices will be column vectors, to avoid the annoying

use of transposes.

2.2 FINANCIAL MARKET ECONOMIES 7

De�nition 2.2.1 A Financial markets equilibrium is a (x�; z�; p�; q�) suchthat

1: x�i 2 argmaxui(xi) s.t.

p�0(xi0 � !i0) = �q�zi; and

p�s(xis � !is) = asz

i; for s = 1; :::S; and furthermore

2:IXi=1

x�i � !is = 0, for any s = 0; 1; :::; S; andIXi=1

z�i

Financial markets equilibrium is the equilibrium concept we shall careabout. This is because i) Arrow-Debreu markets are perhaps too demanding arequirement, and especially because ii) we are interested in �nancial marketsand asset prices q in particular. Arrow-Debreu equilibrium will be a usefulconcept insofar as it represents a benchmark (about which we have a wealthof available results) against which to measure Financial markets equilibrium.

Remark 2.2.2 The economy just introduced is characterized by asset mar-kets in zero net supply, that is, no endowments of assets are allowed for.It is straightforward to extend the analysis to assets in positive net supply,e.g., stocks. In fact, part of each agent i�s endowment (to be speci�c: theprojection of his/her endowment on the asset span, < A >= f� 2 RS : � =Az; z 2 RJg) can be represented as the outcome of an asset endowment, ziw;that is, letting !i1 = (!

i11; :::; !

i1S), we can write

!i1 = wi1 + Az

iw

and proceed straightforwardly by constructing the budget constraints and theequilibrium notion.

No-arbitrage

Before deriving the properties of asset prices in equilibrium, we shall investsome time in understanding the implications that can be derived from themilder condition of no-arbitrage. This is because the characterization of no-arbitrage prices will also be useful to characterize �nancial markets equilbria.For notational convenience, de�ne the (S + 1)� J matrix

W =

24 �qA

35 :


De�nition 2.2.3 W satis�es the No-arbitrage condition if there does notexist a

there does not exist a z 2 RJ such that Wz > 0:5

The No-Arbitrage condition can be equivalently formulated in the follow-ing way. De�ne the span of W to be

< W >= f� 2 RS+1 : � = Wz; z 2 RJg:

This set contains all the feasible wealth transfers, given asset structure A.Now, we can say that W satis�es the No-arbitrage condition if

< W >\RS+1+ = f0g:

Clearly, requiring that W = (�q; A) satis�es the No-arbitrage condition isweaker than requiring that q is an equilibrium price of the economy (withasset structure A). By strong monotonicity of preferences, No-arbitrage isequivalent to requiring the agent�s problem to be well de�ned. The nextresult is remarkable since it provides a foundation for asset pricing basedonly on No-arbitrage.

Theorem 2.2.4 (No-Arbitrage theorem)

< W >\RS+1+ = f0g () 9�̂ 2 RS+1++ such that �̂W = 0:

First, observe that there is no uniqueness claim on the �̂, just existence.Next, notice how �̂W = 0 implies �̂� = 0 for all � 2< W > : It then providesa pricing formula for assets:

�̂W =

0BBB@:::

��̂0qj + �̂1aj1 + :::+ �̂SajS

:::

1CCCA =

0BBB@:::

0

:::

1CCCAJx1

and, rearranging, we obtain for each asset j,

qj = �1aj1 + :::+ �Sa

jS; for �s =

�̂s�̂0

(2.2)

Note how the positivity of all components of �̂ is necessary to obtain (2.2).A few �nal remarks to this section.5Wz > 0 requires that all components of Wz are � 0 and at least one of them > 0:


Remark 2.2.5 An asset which pays one unit of numeraire in state s andnothing in all other states (Arrow security), has price �s according to (2.2).Such asset is called Arrow security.

Remark 2.2.6 Is the vector �̂ obtained by the No-arbitrage theorem unique?Notice how (??) de�nes a system of J equations and S unknowns, representedby �. De�ne the set of solutions to that system as

R(q) = f� 2 RS++ : q = �Ag:

Suppose, the matrix A has rank J 0 � J (that it, A has J 0 linearly independentcolumn vectors and J 0 is the e¤ective dimension of the asset space). Ingeneral, then R(q) will have dimension S � J 0. It follows then that, in thiscase, the No-arbitrage theorem restricts �̂ to lie in a S � J 0 + 1 dimensionalset. If we had S linearly independent assets, the solution set has dimensionzero, and there is a unique � vector that solves (??). The case of S linearlyindependent assets is referred to as Complete markets.

Remark 2.2.7 Let preferences be Von Neumann-Morgernstern:

ui(xi) = ui(xi0) +X

s=1;:::;S

probsui(xis)

whereX

s=1;:::;S

probs = 1: Let then ms =�sprobs

. Then

qj = E (mAj)

In this representation of asset prices the vector m 2 RS++ is called Stochasticdiscount factor.

2.2.1 The stochastic discount factor

In the previous section we showed the existence of a vector that provides thebasis for pricing assets in a way that is compatible with equilibrium, albeitmilder than that. In this section, we will strengthen our assumptions andstudy asset prices in a full-�edged economy. Among other things, this willallow us to provide some economic content to the vector �:Recall the de�nition of Financial market equilibrium. Let MRSis(x

i)denote agent i�s marginal rate of substitution between consumption of the


numeraire good 1 in state s and consumption of the numeraire good 1 atdate 0:

MRSis(xi) =

@ui(xis)

@xi1s

@ui(xi0)

@xi10

Let MRSi(xi) = (: : :MRSis(xi) : : :) denote the vector of marginal rates

of substitution for agent i, an S dimentional vector. Note that, under theassumption of strong monotonicity of preferences, MRSi(xi) 2 RS++:By taking the �rst order conditions (necessary and su¢ cient for a max-

imum under the assumption of strict quasi-concavity of preferences) withrespect to zij of the individual problem for an arbitrary price vector q, weobtain that

qj =SXs=1

probsMRSis(x

i)ajs = E�MRSi(xi) � aj

�; (2.3)

for all j = 1; :::; J and all i = 1; :::; I; where of course the allocation xi is theequilibrium allocation. At equilibrium, therefore, the marginal cost of onemore unit of asset j, qj, is equalized to the marginal valuation of that agentfor the asset�s payo¤,

PSs=1 probsMRS

is(x

i)ajs.Compare equation (2.3) to the previous equation (2.2). Clearly, at any

equilibrium, condition (2.3) has to hold for each agent i. Therefore, in equi-librium, the vector of marginal rates of substitution of any arbitrary agent ican be used to price assets; that is any of the agents�vector of marginal ratesof substitution (normalized by probabilities) is a viable stochastic discountfactor m:In other words, any vector (: : : probsMRSis(x

i) : : :) belongs to R(q) and ishence a viable � for the asset pricing equation (2.2). But recall that R(q) is ofdimension S�J 0; where J 0 is the e¤ective dimension of the asset space. Thehigher the the e¤ective dimension of the asset space (sloppily said, the larger�nancial markets) the more aligned are agents�marginal rates of substitutionat equilibrium (sloppily said, the smaller are unexploited gains from tradeat equilibrium). In the extreme case, when markets are complete (that is,when the rank of A is S), the set R(q) is in fact a singleton and hence theMRSi(xi) are equalized across agents i at equilibrium: MRSi(xi) = MRS;for any i = 1; :::; I:


We conclude that, when markets are Complete, equilibrium allocationsare Pareto optimal. That is, the First Welfare theorem holds for Financialmarket equilibria when markets are Complete.

2.2.2 Arrow theorem

The Arrow theorem is the fondamental decentralization result in �nancialeconomics. It states su¢ cient conditions for a form of equivalence betweenthe Arrow-Debreu and the Financial market equilibrium concepts. It wasessentially introduced by Arrow (1952). The proof of the theorem introducesa reformulation of the budget constraints of the Financial market economywhich focuses on feasible wealth transfers across states directly, on the spanof A,

< A >=�� 2 RS : � = Az; z 2 RJ

in particular. Such a reformulation is important not only in itself but as alemma for welfare analysis in Financial market economies.

Proposition 2.2.8 Let (x�; ��) represent an Arrow-Debreu equilibrium. Sup-pose rank(A) = S (�nancial markets are Complete). Then (x�; z�; p�; q�) isa Financial market equilibrium, where

��s = �sp�s; for any s = 1; :::; S: and

q� =SXs=1

probsMRSis(x

i�)As

Futhermore, the converse also holds: if (x�; z�; p�; q�) is a Financial marketequilibrium of an economy with rank(A) = S, (x�; ��) represents an Arrow-Debreu equilibrium, where ��s = �sp

�s; for any s = 1; :::; S:

A sketch of the argument of the proof is important to understand severalresults in this chapter. Financial market equilibrium prices of assets q� satisfyNo-arbitrage. There exists then a vector �̂ 2 RS+1++ such that �̂W = 0; orq� = �A. The budget constraints in the �nancial market economy are

p�0�xi�0 � !i0

�+ q�zi� = 0

p�s�xi�s � !is

�= Asz

i�; for s = 1; :::S:


Substituting q� = �A; expanding the �rst equation, and writing the con-straints at time 1 in vector form, we obtain:

p�0�xi�0 � !i0

�+

SXs=1

�sp�s

�xi�s � !is

�= 0 (2.4)26666666664

:

:

p�s (xi�s � !is)

:

:

377777777752 < A > (2.5)

If rank(A) = S; it follows that< A >= RS; and the constraint

26666666664

:

:


:

:

377777777752<

A > is never binding. Each agent i�s problem is then subject only to

p�0�xi�0 � !i0

�+

SXs=1

�sp�s

�xi�s � !is

�= 0;

the budget constraint in the Arrow-Debreu economy with

��s = �sp�s; for any s = 1; :::; S:

2.2.3 Aggregation

Agent i�s optimization problem in the de�nition of Financial market equi-librium requires two types of simultaneous decisions. On the one hand, theagent has to deal with the usual consumption decisions i.e., she has to decidehow many units of each good to consume in each state. But she also hasto make �nancial decisions aimed at transferring wealth from one state tothe other. In general, both individual decisions are interrelated: the con-sumption and portfolio allocations of all agents i and the equilibrium prices


for goods and assets are all determined simultaneously from the system ofequations formed by (??) and (??). The �nancial and the real sectors ofthe economy cannot be isolated. Under some special conditions, however,the consumption and portfolio decisions of agents can be separated. Thisis typically very useful when the analysis is centered on �nancial issue. Inorder to concentrate on asset pricing issues, most �nance models deal in factwith 1-good economies, implicitly assuming that the individual �nancial de-cisions and the market clearing conditions in the assets markets determinethe �nancial equilibrium, independently of the individual consumption deci-sions and market clearing in the goods markets; that is independently of thereal equilibrium prices and allocations. In this section we shall identify theconditions under which this can be done without loss of generality. This issometimes called "the problem of aggregation."The idea is the following. If we want equilibrium prices on the spot

markets to be independent of equilibrium on the �nancial markets, thenthe aggregate spot market demand for the L goods in each state s shouldmust depend only on the incomes of the agents in this state (and not inother states) and should be independent of the distribution of income amongagents in this state.

Theorem 2.2.9 Budget Separation. Suppose that each agent i�s prefer-ences are separable across states, identical, homothetic within states, andvon Neumann-Morgenstern; i.e. suppose that there exists an homotheticu : RL ! R such that

ui(xi) = u(xi0) +SXs=1

probsu(xis); for all i = 1; ::; I:

Then equilibrium spot prices p� are independent of asset prices q and of theincome distribution; that is, constant in

n!i 2 RL(S+1)++

��PIi=1 !

i giveno:

This result is a consequence of the fact that the consumer�s maximizationproblem in the de�nition of Financial market equilibrium can be decom-posed into a sequence of spot commodity allocation problems and an incomeallocation problem as follows.The spot commodity allocation problems, given the current and antici-

pated spot prices p = (p0; p1; :::; pS) and an exogenously given stream of


�nancial income yi = (yi0; yi1; :::; y

iS) 2 RS+1++ in units of numeraire, is the

following:

maxxi2RL(S+1)+

ui(xi)

s:t:

p0xi0 = y

i0

psxis = y

is; for s = 1; :::S:

Let the L(S + 1) demand functions be given by xils(p; yi), for l = 1; :::; L;

s = 0; 1; :::S, and de�ne now the indirect utility function for income by

vi(yi; p) = ui(xi(p; yi)):

The Income allocation problem, given prices (p; q); endowments !i, and theasset structure A, is the following:

maxzi2RJ ;yi2RS+1++vi(yi; p)

s:t:

p0!i0 � qzi = yi0

ps!is + asz

i = yis; for s = 1; :::S:

By additive separability across states of the utility, we can break the con-sumption allocation problem into S+1 �spot market�problems, each of whichyields the demands xis(ps; y

is) for each state. Because of identical and ho-

mothetic preferences, then, spot prices p�s for each state s; are determinedindependently of asset prices q and of the distribution of endowments f!igi2I .

Remark 2.2.10 The Budget separation theorem can be interpreted as iden-tifying conditions under which studying a single good economy is without lossof generality. To this end, consider the income allocation problem of agent i,given equilibrium spot prices p� :

maxyi2RS+1++

vi(yi; p�)

s:t:

yi0 = p�0!i0 � qzi

yis = p�s!is + asz

i; for s = 1; :::S


If preferences separable across states, identical, homothetic within states, andvon Neumann-Morgenstern, it is straightforward to show that vi(yi; p�) isidentical across agents i and, seen as a function of yi, it satis�es the as-sumptions we have imposed on ui as a function of xi, in Assumption A.2.Let w0 = p�0!

i0; ws = p

�s!

is; for any s = 1; :::; S; and disregard for notational

simplicity the dependence of vi(yi; p�) on p�: The income allocation problembecomes:

maxyi2RS+1++

v(yi)

s:t:

yi0 � w0 = �qzi

yis � ws = aszi; for s = 1; :::S

which is homeomorphic to any agent i�s optimization problem in the de�-nition of Financial market equilibrium with l = 1. Note that yis gains theinterpretation of agent i�s consumption expenditure in state s, while ws isinterpreted as agent i�s income endowment in state s:

The representative agent theorem

A representative agent is the following theoretical construct.

De�nition 2.2.11 Consider a Financial market equilibrium (x�; z�; p�; q�)of an economy populated by i = 1; :::; I agents with preferences ui : X ! Rand endowments !i: A Representative agent for this economy is an agentwith preferences UR : X ! R and endowment !R such that the Financialmarket equilibrium of an associated economy with the Representative agentas the only agent has prices (p�; q�).

In this section we shall identify assumptions which guarantee that theRepresentative agent construct can be invoked without loss of generality.This assumptions are behind much of the empirical macro/�nance literature.

Theorem 2.2.12 Representative agent. Suppose there exists an homo-thetic u : RL ! R such that

ui(xi) = u(xi0) +

SXs=1

probsu(xis); for all i = 1; ::; I:


Let p� denote equilibrium spot prices. If p�s!is 2< A >; then there exist a

map uR : RS+1 ! R such that:

!R =IXi=1

!is;

UR(x) = uR(y0) +SXs=1

probsuR(ys) where ys = p�

IXi=1

xis; s = 0; 1; :::; S

constitutes a Representative agent.

Since the Representative agent is the only agent in the economy, herconsumption allocation and portfolio at equilibrium,

�x�R; z�R

�; are:

x�R = !R =IXi=1

!i

z�R = 0

If the Representative agent�s preferences can be constructed indepen-dently of the equilibrium of the original economy with I agents, then equilib-rium prices can be read out of the Representative agent�s marginal rates ofsubstitution evaluated at

PIi=1 !

i. SincePI

i=1 !i is exogenously given, equi-

librium prices are obtained without computing the consumption allocationand portfolio for all agents at equilibrium, (x�; z�):The Representative agent theorem, as noted, allows us to obtain equilib-

rium prices without computing the consumption allocation and portfolio forall agents at equilibrium, (x�; z�):Let w =

PIi=1w

i: Under the assumptionsof the Representative agent theorem,

q =

SXs=1

MRSs(w)as; for MRSs(w) =@uR(ws)@ws

@uR(w0)@w0

That is, asset prices can be computed from agents�preferences uR : R ! Rand from the aggregate endowment w: This is called the Lucas� trick forpricing assets.Another interesting but misleading result is the "weak" representative

agent theorem, due to Constantinides (1982).


Theorem 2.2.13 Suppose markets are complete (rank(A) = S) and pref-erences ui(xi) are von Neumann-Morgernstern (but not necessarily identi-cal nor homothetic). Let (x�; z�; p�; q�) be a Financial markets equilibrium.Then,

!R =IXi=1

!i;

UR(x) = max(xi)Ii=1

IXi=1

�iui(xi) s.t.IXi=1

xi = x; where �i = (�i)�1 and �i =

@ui(xi�)

@xi�10

constitutes a Representative agent.

Clearly, then,

q� =SXs=1

MRSRs (!Rs )as;

where MRSRs (x) =@UR(x)@xs

@UR(x)@x0

:

This result is certainly very general, as it does not impose identical ho-mothetic preferences, however, it is not as useful as the �real�Representativeagent theorem to �nd equilibrium asset prices. The reason is that to de�nethe speci�c weights for the planner�s objective function, (�i)Ii=1; we need toknow what the equilibrium allocation, x�; which in turn depends on the wholedistribution of endowments over the agents in the economy.

2.2.4 Asset pricing

Relying on the Aggregation theorem in the previous section, in this sectionwe will abstract from the consumption allocation problems and concentrateon one-good economies. This allows us to simplify the equilibrium de�nitionas follows.Often in �nance, especially in empirical �nance, we study asset pricing

representation which express asset returns in terms of risk factors. Factorsare to be interpreted as those component of the risks that agents do requirea higher return to hold.How do we go from our basic asset pricing equation

q = E(mA)

to factors?


Single-factor beta representation

Consider the basic asset pricing equation for asset j;

qj = E(maj)

Let the return on asset j, Rj, be de�ned as Rj =Ajqj. Then the asset pricing

equation becomes1 = E(mRj)

This equation applied to the risk free rate, Rf , becomes Rf = 1Em. Using the

fact that for two random variables x and y, E(xy) = ExEy + cov(x; y), wecan rewrite the asset pricing equation as:

ERj =1

Em� cov(m;Rj)

Em= Rf � cov(m;Rj)

Em

or, expressed in terms of excess return:

ERj �Rf = �cov(m;Rj)

Em

Finally, letting

�j = �cov(m;Rj)

var(m)

and

�� =var(m)

Em

we have the beta representation of asset prices:

ERj = Rf + �j�m (2.6)

We interpret �j as the "quantity" of risk in asset j and �m (which is thesame for all assets j) as the "price" of risk. Then the expected return ofan asset j is equal to the risk free rate plus the correction for risk, �j�m.Furthermore, we can read (2.6) as a single factor representation for assetprices, where the factor is m, that is, if the representative agent theoremholds, her intertemporal marginal rate of substitution.


Multi-factor beta representations

A multi-factor beta representation for asset returns has the following form:

ERj = Rf +

FXf=1

�jf�mf(2.7)

where (mf )Ff=1 are orthogonal random variables which take the interpretation

of risk factors and

�jf = �cov(mf ; Rj)

var(mf )

is the beta of factor f , the loading of the return on the factor f .

Proposition 2.2.14 A single factor beta representation

ERj = Rf + �j�m

is equivalent to a multi-factor beta representation

ERj = Rf +

FXf=1

�jf�mfwith m =

FXf=1

bfmf

In other words, a multi-factor beta representation for asset returns isconsistent with our basic asset pricing equation when associated to a linearstatistical model for the stochastic discount factor m, in the form of m =PF

f=1 bfmf .

The CAPM

The CAPM is nothing else than a single factor beta representation of thefollowing form:

ERj = Rf + �jf�mf

wheremf = a+ bR

w

the return on the market portfolio, the aggregate portfolio held by the in-vestors in the economy.


It can be easily derived from an equilibrium model under special assump-tions.For example, assume preferences are quadratic:

u(xio; xi1) = �

1

2(xi � x#)2 � 1

2�

SXs=1

probs(xis � x#)2

Moreover, assume agents have no endowments at time t = 1. LetPI

i=1 xis =

xs; s = 0; 1; :::; S; andPI

i=1wi0 = w0. Then budget constraints include

xs = Rws (w0 � x0)

Then,

ms = �xs � x#x0 � x#

=�(w0 � x0)(x0 � x#)

Rws ��x#

x0 � x#

which is the CAPM for a = � �x#

x0�x# and b =�(w0�x0)(x0�x#) :

Note however that a = �x#

x0�x# and b =�(w0�x0)(x0�x#) are not constant, as they

do depend on equilibrium allocations. This will be important when we studyconditional asset market representations, as it implies that the CAPM isintrinsically a conditional model of asset prices.

Bounds on stochastic discount factors

Write the beta representation of asset returns as:

ERj �Rf = cov(m;Rj)

Em=�(m;Rj)�(m)�(Rj)

Em

where 0 � �(m;Rj) � 1 denotes the correlation coe¢ cient and �(:), thestandard deviation. Then

j ERj �Rf�(Rj)

j� �(m)

Em

The left-hand-side is the Sharpe-ratio of asset j.The relationship implies a lower bound on the standard deviation of any

stochastic discount factor m which prices asset j. Hansen-Jagannathan areresponsible for having derived bounds like these and shown that, when thestochastic discount factor is assumed to be the intertemporal marginal rate


of substitution of the representative agent (with CES preferences), the datadoes not display enough variation in m to satisfy the relationship.

A related bound is derived by noticing that no-arbitrage implies the ex-istence of a unique stochastic discount factor in the space of asset payo¤s,denoted mp, with the property that any other stochastic discount factor msatis�es:

m = mp + �

where � is orthogonal to mp.The following corollary of the No-arbitrage theorem leads us to this result.

Corollary 2.2.15 Let (A; q) satisfy No-arbitrage. Then, there exists a unique� � 2< A > such that q = A� �:

We can now exploit this uniqueness result to yield a characterization ofthe �multiplicity�of stochastic discount factors when markets are incomplete,and consequently a bound on �(m). In particular, we show that, for a given(q; A) pair a vector m is a stochastic discount factor if and only if it canbe decomposed as a projection on < A > and a vector-speci�c componentorthogonal to < A >. Moreover, the previous corollary states that such aprojection is unique.

Let m 2 RS++ be any stochastic discount factor, that is, for any s =1; : : : ; S, ms =

�sprobs

and qj = E(mAj); for j = 1; :::; J: Consider the orthogo-nal projection of m onto < A >, and denote it by mp. We can then write anystochastic discount factors m as m = mp + ", where " is orthogonal to anyvector in < A >; in particular to any Aj. Observe in fact that mp+ " is alsoa stochastic discount factors since qj = E((mp+")aj) = E(mpaj)+E("aj) =E(mpaj), by de�nition of ". Now, observe that qj = E(mpaj) and that wejust proved the uniqueness of the stochastic discount factors lying in < A > :In words, even though there is a multiplicity of stochastic discount factors,they all share the same projection on < A >. Moreover, if we make the eco-nomic interpretation that the components of the stochastic discount factorsvector are marginal rates of substitution of agents in the economy, we caninterpret mp to be the economy�s aggregate risk and each agents " to be theindividual�s unhedgeable risk.It is clear then that

�(m) � �(mp)

the bound on �(m) we set out to �nd.


2.2.5 Pareto optimality

Under Complete markets, the First Welfare Theorem holds for a Financialmarket equilibrium. This is a direct implication of Arrow theorem.

Proposition 2.2.16 Let (x�; z�; p�; q�) be a Financial market equilibrium ofan economy with Complete markets (with rank(A) = S): Then x� is a Paretooptimal allocation.

However, under Incomplete markets Financial market equilibria are gener-ically ine¢ cient in a Pareto sense. That is, a planner could �nd an allocationthat improves some agents without making any other agent worse o¤.

Theorem 2.2.17 At a Financial Market Equilibrium (x�; z�; p�; q�) of anincomplete �nancial market economy, that is, of an economy with rank(A) <S, the allocation x� is generically6 not Pareto Optimal.7

Pareto optimality might however represent too strict a de�nition of socialwelfare of an economy with frictions which restrict the consumption set, as inthe case of incomplete markets. In this case, markets are assumed incompleteexogenously. There is no reason in the fundamentals of the model why theyshould be, but they are. Under Pareto optimality, however, the social welfarenotion does not face the same contraints. For this reason, we typically de�ne aweaker notion of social welfare, Constrained Pareto optimality, by restrictingthe set of feasible allocations to satisfy the same set of constraints on theconsumption set imposed on agents at equilibrium. In the case of incompletemarkets, for instance, the feasible wealth vectors across states are restrictedto lie in the span of the payo¤ matrix. That can be interpreted as theeconomy�s ��nancial technology�and it seems reasonable to impose the sametechnological restrictions on the planner�s reallocations. The formalization

6We say that a statement holds generically when it holds for a full Lebesgue-measuresubset of the parameter set which characterizes the economy. In these notes we shallassume that the an economy is parametrized by the endowments for each agent, the assetpayo¤ matrix, and a two-parameter parametrization of utility functions for each agent;see Magill-Shafer, ch. 30 in W. Hildenbrand and H. Sonnenschein (eds.), Handbook ofMathematical Economics, Vol. IV, Elsevier, 1991.

7The proof can be found in Magill-Shafer, ch. 30 in W. Hildenbrand and H. Sonnen-schein (eds.), Handbook of Mathematical Economics, Vol. IV, Elsevier, 1991. It requiresmathematical tecniques from di¤erential topology which are not appropriate to be intro-duced in this course.


of an e¢ ciency notion capturing this idea follows. Let xit=1 = (xis)Ss=1 2 RSL+ ;

and similarly pt=1 = (ps)Ss=1 2 RSL+

De�nition 2.2.18 (Diamond, 1968; Geanakoplos-Polemarchakis, 1986) Let(x�; z�; p�; q�) represent a Financial market equilibrium of an economy whoseconsumption set at time t = 1 is restricted by

xit=1;2 B(pt=1); for any i = 1; :::; IIn this economy, the allocation x� is Constrained Pareto optimal if there doesnot exist a (y; �) such that

1: u(yi) � u(x�i) for any i = 1; :::; I, strictly for at least one i

2:IXi=1

yis � !is = 0, for any s = 0; 1; :::; S

and3: yit=1 2 B(g�t=1(!; �)); for any i = 1; :::; I

where g�t=1(!; �) is a vector of equilibrium prices for spot markets at t = 1opened after each agent i = 1; :::; I has received income transfer A�i:

The constraint on the consumption set restricts only time 1 consump-tion allocations. More general constraints are possible but these formulationis consistent with the typical frictions we encounter in economics, e.g., on�nancial markets. It is important that the constraint on the consumptionset depends in general on g�t=1(!; �), that is on equilibrium prices for spotmarkets opened at t = 1 after income transfers to agents. It implicit identi-�es income transfers (besides consumption allocations at time t = 0) as theinstrument available for Constrained Pareto optimality; that is, it implicitlyconstrains the planner implementing Constraint Pareto optimal allocationsto interact with markets, speci�cally to open spot markets after transfers.On the other hand, the planner is able to anticipate the spot price equilib-rium map, g�t=1(!; �); that is, to internalize the e¤ects of di¤erent transferson spot prices at equilibrium.

Proposition 2.2.19 Let (x�; z�; p�; q�) represent a Financial market equi-librium of an economy with complete markets (rank(A) = S) and whoseconsumption set at time t = 1 is restricted by

xit=1 2 B � RSL+ ; for any i = 1; :::; IIn this economy, the allocation x� is Constrained Pareto optimal.


Crucially, markets are Complete and B is independent of prices. Theproof is then a straightforward extension of the First Welfare theorem com-bined with Arrow theorem. Constraint Pareto optimality of Financial marketequilibrium allocations is guaranteed as long as the constraint set B is ex-ogenous.

Proposition 2.2.20 Let (x�; z�; p�; q�) represent a Financial market equi-librium of an economy with Incomplete markets (rank(A) < S). In thiseconomy, the allocation x� is not Constrained Pareto optimal.

There is a fundamental di¤erence between incomplete market economies,which have typically not Constrained Optimal equilibrium allocations, andeconomies with constraints on the consumption set, which have, on the con-trary, Constrained Optimal equilibrium allocations. It stands out by com-paring the respective trading constraints

g�s(!s; �)(xis � !is) = As�i; for all i and s, vs. xit=1 2 B, for all i:

The trading constraint of the incomplete market economy is determined atequilibrium, while the constraint on the consumption set is exogenous. An-other way to re-phrase the same point is the following. A planner choosing(y; �) will take into account that at each (y; �) is typically associated a dif-ferent trading constraint g�s(!s; �)(x

is � !is) = As�i; for all i and s; while any

agent i will choose (xi; zi) to satisfy p�s(xis � !is) = Aszi; for all s, taking as

given the equilibrium prices p�s:The constrained ine¢ ciency due the dependence of constraints on equilib-

rium prices is sometimes called a pecuniary externality.8 Several examples ofsuch form of externality/ine¢ ciency have been developed recently in macro-economics. Some examples are: Thomas (1995), Krishnamurthy (2003), Ca-ballero and Krishnamurthy (2003), Lorenzoni (2008). Kocherlakota (2009),Davila, Hong, Krusell, and Rios Rull (2005).

Remark 2.2.21 Consider an economy whose constraints on the consump-tion set depend on the equilibrium allocation:

xit=1 2 B(x�t=1; z�); for any i = 1; :::; I

This is essentially an externality in the consumption set. It is not hard toextend the analysis of this section to show that this formulation introducesine¢ ciencies and equilibrium allocations are Constraint Pareto sub-optimal.

8The name is due to Joe Stiglitz (or is it Greenwald-Stiglitz?).

2.3 CORPORATE FINANCE ECONOMIES 25

Corollary 2.2.22 Let (x�; z�; p�; q�) represent a Financial market equilib-rium of a 1-good economy (L = 1) with Incomplete markets (rank(A) < S).In this economy, the allocation x� is Constrained Pareto optimal.

In fact, the constraint on the consumption set implied by incompletemarkets, if L = 1, can be written

(xis � !is) = Aszi;

and it is hence ndependent of prices, of the form xit=1 2 B.

Remark 2.2.23 Consider an alternative de�nition of Constrained Paretooptimality, due to Grossman (1970), in which constraints 3 are substitutedby

30:

26666666664

:

:


:

:

37777777775= Azi; for any i = 1; :::; I

where p� is the spot market Financial market equilibrium vector of prices.That is, the planner takes the equilibrium prices as given. It is immediate toprove that, with this de�nition of Constrained Pareto optimality, any Finan-cial market equilibrium allocation x�of an economy with Incomplete marketsis in fact Constrained Pareto optimal, independently of the �nancial marketsavailable (rank(A) � S):

Remark 2.2.24 Consider a 1-good (L = 1) Incomplete market economy(rank(A) < S) which lasts 3 periods. Note that Financial market equilibriumallocations of such an economy are not Constrained Pareto optimal.

2.3 Corporate �nance economies

Assume for simplicity that L = 1, and that there is a single type of �rm in theeconomy which produces the good at date 1 using as only input the amount


k of the commodity invested in capital at time 0:9 The output depends onk according to the function f(k; s), de�ned for k 2 K, where s is the staterealized at t = 1. We assume that

- f(k; s) is continuously di¤erentiable, increasing and concave in k;

- �; K are closed, compact subsets of R+ and 0 2 K.

In addition to �rms, there are I types of consumers. The demand side ofthe economy is as in the previous section, except that each agent i 2 I is alsoendowed with �i0 units of stock of the representative �rm. Consumer i hasvon Neumann-Morgernstern preferences over consumption in the two dates,represented by ui (xi0) + Eui (xi), where ui (�) is continuously di¤erentiable,strictly increasing and strictly concave.Let the outstanding amount of equity be normalized to 1: the initial

distribution of equity among consumers satis�esP

i �i0 = 1. The problem of

the �rm consists in the choice of its production plan k:.Firms are perfectly competitive and hence take prices as given. The

�rm�s cash �ow, f(k; s); varies with k. Thus equity is a di¤erent �product�for di¤erent choices of the �rm. What should be its price when all thiscontinuum of di¤erent �products�are not actually traded in the market? Inthis case the price is only a �conjecture.�It can be described by a map Q(k)specifying the market valuation of the �rm�s cash �ow for any possible valueof its choice k.10 The �rm chooses its production plan k so as to maximizeits value. The �rm�s problem is then:

maxk�k +Q(k) (2.8)

When �nancial markets are complete, the present discounted valuation ofany future payo¤ is uniquely determined by the price of the existing assets.This is no longer true when markets are incomplete, in which case the pricesof the existing assets do not allow to determine unambiguously the value ofany future cash �ow. The speci�cation of the price conjecture is thus moreproblematic in such case. Let k� denote the solution to this problem.

9It should be clear from the analysis which follows that our results hold unalteredif the �rms� technology were described, more generally, by a production possibility setY � RS+1.10These price maps are also called price perceptions.


At t = 0, each consumer i chooses his portfolio of �nancial assets and ofequity, zi and �i respectively, so as to maximize his utility, taking as giventhe price of assets, q and the price of equity Q. In the present environment aconsumer�s long position in equity identi�es a �rm�s equity holder, who mayhave a voice in the �rm�s decisions. It should then be treated as conceptuallydi¤erent from a short position in equity, which is not simply a negativeholding of equity. To begin with, we rule out altogether the possibility ofshort sales and assume that agents can not short-sell the �rm equity:

�i � 0; 8i (2.9)

The problem of agent i is then:

maxxi0;x

i;zi;�iui�xi0�+ Eui

�xi�

(2.10)

subject to (2.9) and

xi0 = !i0 + [�k +Q] �i0 �Q�i � q zi (2.11)

xi(s) = !i(s) + f(k; s)�i + A(s)zi; 8s 2 S (2.12)

Let�xi�0 ; x

i�; zi�; �i��denote the solution to this problem.

In equilibrium, the following market clearing conditions must hold, forthe consumption good:11X

i

xi0 + k �Xi

!i0Xi

xi(s) �Xi

!i(s) + f(k; s); 8s 2 S

or, equivalently, for the assets: Xi

zi = 0 (2.13)Xi

�i = 1 (2.14)

11We state here the conditions for the case of symmetric equilibria, where all �rms takethe same production and �nancing decision, so that only one type of equity is availablefor trade to consumers. They can however be easily extended to the case of asymmetricequilibria as, for instance, in the example of Section ??.


In addition, the equity price map faced by �rms must satisfy the followingconsistency condition:

i) Q(k�) = Q;

This condition requires that, at equilibrium, the price of equity conjec-tured by �rms coincides with the price of equity, faced by consumers in themarket: �rms�conjectures are �correct�in equilibrium.We also restrict out of equilibrium conjectures by �rms, requiring that

they satisfy:

ii) Q(k) = maxi E [MRSi�f(k)], 8k, whereMRSi� denotes the marginal rateof substitution between consumption at date 0 and at date 1 in states for consumer i; evaluated at his equilibrium consumption allocation(xi�0 ; x

i�).

Condition ii) says that for any k (not just at equilibrium!) the value ofthe equity price map Q(k) equals the highest marginal valuation - across allconsumers in the economy - of the cash �ow associated to k. The consumers�marginal rates of substitutions MRS

i(s) used to determine the market val-

uation of the future cash �ow of a �rm are taken as given, una¤ected bythe �rm�s choice of k. This is the sense in which, in our economy, �rms arecompetitive: each �rm is �small�relative to the mass of consumers and eachconsumers holds a negligible amount of shares of the �rm.To better understand the meaning of condition ii), note that the con-

sumers with the highest marginal valuation for the �rm�s cash �ow whenthe �rm chooses k are those willing to pay the most for the �rm�s equity inthat case and the only ones willing to buy equity - at the margin - whenits price satis�es ii). Given i) such property is clearly satis�ed for the �rms�equilibrium choice k�. Condition ii) requires that the same is true for anyother possible choice k: the value attributed to equity equals the maximumany consumer is willing to pay for it. Note that this would be the equilibriumprice of equity of a �rm who were to �deviate�from the equilibrium choiceand choose k instead: the supply of equity with cash �ow corresponding tok is negligible and, at such price, so is its demand.In this sense, we can say that condition ii) imposes a consistency con-

dition on the out of equilibrium values of the equity price map; that is, itcorresponds to a "re�nement" of the equilibrium map, somewhat analogous


to bacward induction. Equivalently, when price conjectures satisfy this con-dition, the model is equivalent to one where markets for all the possible typesof equity (that is, equity of �rms with all possible values of k) are open, avail-able for trade to consumers and, in equilibrium all such markets - except theone corresponding to k� - clear at zero trade.12

It readily follows from the consumers��rst order conditions that in equi-librium the price of equity and of the �nancial assets satisfy:

Q = maxiE�MRSi� � f(k�)

�(2.15)

q = E�MRSi� � A

�The de�nition of competitive equilibrium is stated for simplicity for the

case of symmetric equilibria, where all �rms choose the same productionplan. When the equity price map satis�es the consistency conditions i) andii) the �rms�choice problem is not convex. Symmetric equilibria are thennot guaranteed to exist, and asymmetric equilibria might obtain, in whichdi¤erent �rms choose di¤erent production plans.Starting with the initial contributions of Diamond (1967), Dreze (1974),

Grossman-Hart (1979), and Du¢ e-Shafer (1986), a large literature has dealtwith the question of what is the appropriate objective function of the �rmwhen markets are incomplete.The issue arises because, as mentioned above,�rms�production decisions may a¤ect the set of insurance possibilities avail-able to consumers by trading in the asset markets.If agents are allowed in�nite short sales of the equity of �rms, as in the

standard incomplete market model, a small �rm will possibly have a largee¤ect on the economy by choosing a production plan with cash �ows which,when traded as equity, change the asset span. It is clear that the pricetaking assumption appears hard to justify in this context, since changes inthe �rm�s production plan have non-negligible e¤ects on allocations and henceequilibrium prices. The incomplete market literature has struggled with thisissue, trying to maintain a competitive equilibrium notion in an economicenvironment in which �rms are potentially large.In the environment considered in these notes, this problem is avoided

by assuming that consumers face a constraint preventing short sales, (2.9),

12An analogous speci�cation of the price conjecture has been earlier considered byMakowski (1980) and Makowski-Ostroy (1987) in a competitive equilibrium model withdi¤erentiated products, and by Allen-Gale (1991) and Pesendorfer (1995) in models of�nancial innovation.


which guarantees that each �rm�s production plan has instead a negligible(in�nitesimal) e¤ect on the set of admissible trades and allocations availableto consumers. Evidently, for price taking behavior to be justi�ed a no shortsale constraint is more restrictive than necessary and a bound on short salesof equity would su¢ ce; see Bisin-Gottardi-Ruta (2010).When short sales are not allowed, the decisions of a �rm have a negligible

e¤ect on equilibrium allocations and market prices. However, each �rm�s de-cision has a non-negligible impact on its present and future cash �ows. Pricetaking can not therefore mean that the price of its equity is taken as givenby a �rm, independently of its decisions. However, as argued in the previoussection, the level of the equity price associated to out-of-equilibrium valuesof k is not observed in the market. It is rather conjectured by the �rm. In acompetitive environment we require such conjecture to be consistent, as re-quired by condition ii) in the previous section. This notion of consistency ofconjectures implicitly requires that they be competitive, that is, determinedby a given pricing kernel, independent of the �rm�s decisions.13 But whichpricing kernel? Here lies the core of the problem with the de�nition of the ob-jective function of the �rm when markets are incomplete. When markets areincomplete, in fact, the marginal valuation of out-of-equilibrium productionplans di¤ers across di¤erent agents at equilibrium. In other words, equityholders are not unanimous with respect to their preferred production planfor the �rm. The problem with the de�nition of the objective function of the�rm when markets are incomplete is therefore the problem of aggregatingequity holders�marginal valuations for out-of-equilibrium production plans.The di¤erent equilibrium notions we �nd in the literature di¤er primarily inthe speci�cation of a consistency condition on Q (k), the price map whichthe �rms adopts to aggregate across agents�marginal valuations.14

Consider for example the consistency condition proposed by Dreze (1974):

QD(k) = E

"Xi

�i�MRSi�f(k)

#; 8k (2.16)

13Independence of the kernel is guarantee by the fact that MRSi�(s); for any i, isevaluated at equilibrium.14A minimal consistency condition on Q (k) is clearly given by i) in the previous section,

which only requires the conjecture to be correct in correspondence to the �rm�s equilibriumchoice. Du¢ e-Shafer (1986) indeed only impose such condition and �nd a rather largeindeterminacy of the set of competitive equilibria.


Such condition requires the price conjecture for any plan k to equal thepro rata marginal valuation of the agents who at equilibrium are the �rm�sequity holders (that is, the agents who value the most the plan chosen by�rms in equilibrium). It does not however require that the �rm�s equityholders are those who value the most any possible plan of the �rm, withoutcontemplating the possibility of selling the �rm in the market, to allow thenew equity buyers to operate the production plan they prefer. Equivalently,the value of equity for out of equilibrium production plans is determined usingthe - possibly incorrect - conjecture that the �rms�equilibrium shareholderswill still own the �rm out of equilibrium.Grossman-Hart (1979) propose another consistency condition and hence

a di¤erent equilibrium notion. In their case

QGH(k) = E

"Xi

�i0MRSi�f(k)

#; 8k

We can interpret such notion as describing a situation where the �rm�s planis chosen by the initial equity holders (i.e., those with some predeterminedstock holdings at time 0) so as to maximize their welfare, again withoutcontemplating the possibility of selling the equity to other consumers whovalue it more. Equivalently, the value of equity for out of equilibrium pro-duction plans is again derived using the conjecture belief that �rms�initialshareholders stay in control of the �rm out of equilibrium.

Pareto optimality

A consumption allocation (xi0; xi)Ii=1 is admissible if:

15

1. it is feasible: there exists a production plan k such thatXi

xi0 + k �Xi

!i0 (2.17)Xi

xi(s) �Xi

!i(s) + f(k; s); 8s 2 S (2.18)

15To keep the notation simple, we state both the de�nition of competitive equilibria andadmissible allocations for the case of symmetric allocations. The analysis, including thee¢ ciency result ,extends however to the case where asymmetric allocations are allowed areadmissible; see also the next section.


2. it is attainable with the existing asset structure: for each consumer i,there exists a pair

�zi; �i

�such that:

xi(s) = !i(s) + f(k; s) �i + A(s)zi; 8s 2 S (2.19)

Next we present the notion of e¢ ciency restricted by the admissibilityconstraints:

De�nition 2.3.1 A competitive equilibrium allocation is constrained Paretoe¢ cient if we can not �nd another admissible allocation which is Paretoimproving.

The validity of the First Welfare Theorem with respect to such notioncan then be established by an argument essentially analogous to the one usedto establish the Pareto e¢ ciency of competitive equilibria in Arrow-Debreueconomies.

Theorem 2.3.2 (First Welfare) Competitive equilibria are constrainedPareto e¢ cient.

Unanimity

Under the de�nition of equilibrium proposed in these notes, equity holdersunanimously support the �rm�s choice of the production and �nancial deci-sions which maximize its value (or pro�ts), as in (2.8). This follows from thefact that, when the equity price map satis�es the consistency conditions i)and ii), the model is equivalent to one where a continuum of types of equityis available for trade to consumers, corresponding to any possible choice ofk the representative �rm can make, at the price Q(k). Thus, for any pos-sible value of k a market is open where equity with a payo¤ f(k; s) can betraded, and in equilibrium such market clears with a zero level of trades forthe values of k not chosen by the �rms.For any possible choice k of a �rm, the (marginal) valuation of the �rm

by an agent i isE�MRSi� � f(k�)

�;

and it is always weakly to the market value of the �rm, given by

maxiE�MRSi� � f(k�)

�:


Proposition 2.3.3 At a competitive equilibrium, equity holders unanimouslysupport the production k�; that is, every agent i holding a positive initialamount �i0 of equity of the representative �rm will be made - weakly - worseo¤ by any other choice k0 of the �rm.

Modigliani-Miller theorem

We examine now the case where �rms take both production and �nancialdecisions, and equity and debt are the only assets they can �nance theirproduction with. The choice of a �rm�s capital structure is given by thedecision concerning the amount B of bonds issued. The problem of the �rmconsists in the choice of its production plan k and its �nancial structure B.To begin with, we assume without loss of generality that all �rms�debt is riskfree. The �rm�s cash �ow in this context is then [f(k; s)�B] and varies withthe �rm�s production and �nancing choices, k;B. Equity price conjectureshave the form Q(k;B), while the price of the (risk free) bond is independentof (k;B); we denote it p. The �rm�s problem is then:

maxk;B

�k +Q(k;B) + p B (2.20)

The consumption side of the economy is the same as in the previoussection, except that now agents can also trade the bond. Let bi denote thebond portfolio of agent i; and let continue to impose no-short sales contraints:

�i � 0

bi � 0; 8i:

Proceeding as in the previous section, at equilibrium we shall require that

Q(k) = maxiE�MRSi� � [f(k)�B]

�;8k;

p = maxiE�MRSi�

�whereMRSi� denotes the marginal rate of substitution between consumptionat date 0 and at date 1 in state s for consumer i; evaluated at his equilibriumconsumption allocation (xi�0 ; x

i�). Suppose now that �nancial markets arecomplete, that is rank(A) = S: At equilibrium then MRSi� = MRS�, 8i.Therefore, in this case

Q(k;B) + p B = E [MRS� � f(k)] ;8k;


and the value of the �rm, Q(k;B) + p B; is independent of B: This provesthe celebrated

Modigliani-Miller theorem. If �nancial markets are complete the �nanc-ing decision of the �rm, B; is indeterminate.

It should be clear that when �nancial markets are not complete and agentsare restricted by no-short sales constraints, the Modigliani-Miller theoremdoes not quite necessarily hold.

2.4 Asymmetric information economies

Do competitive insurance markets function orderly in the presence of moralhazard and adverse selection? What are the properties of allocations at-tainable as competitive equilibria of such economies? And in particular, arecompetitive equilibria incentive e¢ cient?For such economies the interaction between the private information di-

mension (e.g., the unobservable action in the moral hazard case, the unob-servable type in the adverse selection case) and the observability of agents�trades plays a crucial role, since trades have typically informational con-tent over the agents�private information. In particular, to decentralize in-centive e¢ cient Pareto optimal allocations the availability of fully exclusivecontracts, i.e., of contracts whose terms (price and payo¤) depend on thetransactions in all other markets of the agent trading the contract, is gener-ally required. The implementation of these contracts imposes typically thevery strong informational requirement that all trades of an agent need to beobserved.16

The fundamental contribution on competitive markets for insurance con-tracts is Prescott and Townsend (1984a); see also Prescott and Townsend(1984b). They analyze Walrasian equilibria of economies with moral hazardand with adverse selection when exclusive contracts are enforceable, that is,when trades are fully observable.17 In these notes we concentrate on the sim-

16In the context of these economies M. Harris and R. Townsend (1981) prove a versionof the Revelation principle.17The standard strategic analysis of competition in insurance economies, due to

Rothschild-Stiglitz (1976), considers the Nash equilibria of a game in which insurancecompanies simultaneously choose the contracts they issue, and the competitive aspect ofthe market is captured by allowing the free entry of insurance companies. Such equi-

2.4 ASYMMETRIC INFORMATION ECONOMIES 35

pler case of moral hazard. We refer to Bisin-Gottardi (2006) for the case ofadverse selection.18

2.4.1 Moral hazard insurance economies

Agents live two periods, t = 0; 1; and consume a single consumption goodonly in period 1. Uncertainty is purely idiosyncratic and all agents are ex-ante identical. In particular, each agent faces a (date 1) endowment whichis an identically and independently distributed random variable ! on a �nitesupport S.19

Moral hazard (hidden action) is captured by the assumption that theprobability distribution of the period 1 endowment that each agent facesdepends from the value taken by a variable e 2 E, an unobservable level ofe¤ort which is chosen by the agent.Let �s(e) be the probability of the realization !s given e. ObviouslyPs2S �s(e) = 1; for any e 2 E; a compact, convex set: By the Law of

Large Numbers,�s(e) is also the fraction of agents who have chosen e¤ort efor which state s is realized. Agents�preferences are represented by a vonNeumann-Morgenstern utility function of the following form:X

s2S�s(e)u(xs) � v(e)

where v(e) denotes the disutility of e¤ort e. We assume the followingregularity conditions:The utility function u(x) is strictly increasing, strictly concave, twice

continuously di¤erentiable, and limx!0 u0(x) = 1: The cost function v(e)

librium concept does not perform too well: equilibria in pure strategies do not exist forrobust examples (Rothschild-Stiglitz (1976)), while equilibria in mixed strategies exist(Dasgupta-Maskin (1986)) but, in this set-up, are of di¢ cult interpretation. Even whenequilibria in pure strategies do exist, it is not clear that the way the game is modelled isappropriate for such markets, since it does not allow for dynamic reactions to new contracto¤ers (Wilson (1977) and Riley (1979); see also Maskin-Tirole (1992)). Moreover, oncesequences of moves are allowed, equilibria are not robust to �minor�perturbations of theextensive form of the game (Hellwig (1987)).18Other references include Bennardo and Chiappori (2003), Bisin and Gottardi (1999,

2006), Dubey, Geanakoplos and Shubik (2004), Gale (1992, 1996).19Measurability issues arise in probability spaces with a continuum of indipendent ran-

dom variables. We adopt the usual abuse of the Law of Large Numbers.


is strictly increasing, strictly convex, twice continuously di¤erentiable, andsupe2E v

0(e) =1:Essentially without loss of generality, let the state space S be ordered so

that !s > !s�1; for all s = 2; :::; S: We then impose the following standardrestriction.

Single-crossing property. The odds ratio �s(e)�s�1(e)

is strictly increasing ine, for any s = 2; :::; S:

The symmetric information benchmark

Consider now the benchmark case of symmetric information, in which e iscommonly observed. An allocation (x; e) 2 RS+�E of consumption and e¤ortis optimal under symmetric information if it solves:

maxx;e

Xs2S

�s(e)u(xs) � v(e); (2.21)

s.t. Xs2S

�s(e)(xs � !s) = 0

Let qs(e) denote the (linear) price of consumption in state s for agentswho chose e¤ort e. By allowing the prices of the securities whose payo¤ iscontingent on the idiosyncratic uncertainty to depend on e, we e¤ectivelyare introducing price conjectures: we read qs(e) as the price of consumptioncontingent to state s if the agent chooses e¤ort e, for any e 2 E; not justat the equilibrium e. This is the same problem we found in productioneconomies with incomplete markets, where the price faced by the �rm was awhole map Q(k); interpreted as a price conjecture.At a competitive equilibrium each agent solves

maxx;e

Xs2S

�s(e)u(xs) � v(e); (2.22)

s.t. Xs2S

qs(e)(xs � !s) = 0

and markets clear Xs2S

�s(e)(xs � !s) = 0 (2.23)


We impose the following consistency condition on the price conjecture:

qs(e) = �s(e):

Under the consistency condition it is now straightforward to prove theFirst and Second Welfare theorems for this economy under symmetric infor-mation.

Moral hazard

Consider now the case of asymmetric information, in which his choice ofe¤ort e is private information of each agent. In this context, an allocation(x; e) 2 RS+�E of consumption and e¤ort is incentive constrained optimal ifit solves:

maxx;e

Xs2S

�s(e)u(xs) � v(e); (2.24)

s.t. Xs2S

�s(e)(xs � !s) = 0

and

e 2 e(x) = argmaxXs2S

�s(e)u(xs) � v(e); given x

The last constraint, called incentive constraint, requires that the allocation(x; e) must be such that the agent prefers (x; e) to any other allocation (x; e0),for any e; e0 2 E.At a Prescott-Townsend competitive equilibrium prices cannot have the

form qs(e), as the agent�s choice for e is not observed. What is observed,however is the consumption allocation x demanded by the agent in the mar-ket. (Note that the exclusivity assumption guarantees that this is the case,that x is observable). Price conjectures can then be de�ned as a function ofx as

qs(x) = qs(e(x)):

At a competitive equilibrium then each agent solves

maxx;e

Xs2S

�s(e)u(xs) � v(e); (2.25)


s.t. Xs2S

qs(e(x))(xs � !s) = 0


�s(e)(xs � !s) = 0 (2.26)

At an equilibrium (x�; e�), we impose the following consistency conditionon the price conjecture:

qs(e(x)) = �s(e(x))

The First and Second Welfare theorems, in its incentive constrained versions,hold straightforwardly for the moral hazard economy.

Remark 2.4.1 An equivalent notion of equilibrium is possible, which is equiv-alent to the one just proposed. Suppose at a competitive equilibrium eachagent solves

maxx;e

Xs2S

�s(e)u(xs) � v(e); (2.27)

s.t. Xs2S

qs(e)(xs � !s) = 0; andXs2S

�s(e)u(xs) � v(e) �Xs2S

�s(e0)u(xs) � v(e0); for any e0 2 E


�s(e)(xs � !s) = 0 (2.28)

Prices depend on e¤ort e; q(e): As e¤ort is not observed, this is to be inter-preted that prices depend on the (implicit) declaration of e¤ort on the part ofthe agent; for instance di¤erent markets could be present with di¤erent prices- and by choosing one of these where to trade each agent implicitly declareshis e¤ort choice. The conditionX

s2S�s(e)u(xs) � v(e) �

Xs2S

�s(e0)u(xs) � v(e0); for any e0 2 E

ther requires that in the market with prices q(e) only incentive compatible al-locations are o¤ered, that is, only allocations which induce the agent to choose


e¤ort e: It is straightforward to see that this equilibrium notion is equivalentto the one with rational price conjectures we have proposed. The interpre-tation is di¤erent however: with rational conjectures it is the conjectureson prices of non incentive compatible allocations which are restricted, whilewith the equilibrium notion in this remark it is tradable allocations which arerestricted to exclude non incentive compatible ones.

As we noted, the exclusivity assumption guarantees that x is observable.Suppose it is not (this is the non-exclusivity case). The next problem dealswith this case in a simple but instructive example.

2.4.2 Corporate agency economies

[...Bisin, Gottardi, Ruta...]

Chapter 3

In�nite-horizon economies

Note that, when economies have an in�nite horizon, their commodity spaceis in�nite dimensional. Good discussion of the spaces we typically study isin Lucas-Stokey with Prescott (1989); see also Zame (.).Assume a representative-agent economy with one good. Let time be in-

dexed by t = 0; 1; 2; :::: Uncertainty is captured by a probability space repre-sented by a tree. Suppose that there is no uncertainty at time 0 and call s0

the root of the tree. Without much loos of generality, we assume that eachnode has a constant number of successors, S. At generic node at time t iscalled st 2 St. Note that the dimensionality of St increases exponentiallywith time t (abusing notation it 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,where xt : St �! R+ is a random variable on the underlying probabilityspace, for each t. Similarly, let ! := f!tg1t=0 be a stochastic processes de-scribing an agent�s endowments. Let 0 < � < 1 denote the discount factor.

3.1 Arrow-Debreu economies

Suppose that at time zero, the agent can trade in contingent commodities.Let p := fptg1t=0 denote the stochastic process for prices, where pt : St �!R+, for each t.Then ((x�i)i ; p

�) is an Arrow-Debreu Equilibrium if

41

42 CHAPTER 3 INFINITE-HORIZON ECONOMIES

i. given p�;

x�i 2 argmaxfu(x0) + E0[P1

t=1 �tu(xt)]g

s:t:P1t=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, thatis, the whole sequence of x(st) for any st 2 St and any t � 0.

3.2 Financial markets economies

Suppose that throughout the uncertainty tree, there are J assets. We shallallow assets to be long-lived. In fact we shall assume they are and let thereader take care of the straightforward extension in which some of the assetspay o¤ only in a �nite set of future times. Let z := fztg1t=1 denote the se-quence of portfolios of the representative agent, where zt : St �! RJ . Assets�payo¤s are captured at each time t by the S � J matrix At. Furthermore,capital gains are qt � qt�1, and returns are Rt = At+qt

qt�1.

In a �nancial market economy agents do not trade at time 0 only. Theyin fact, at each node st receive endowments and payo¤s from the portfoliosthey carry from the previous node, they re-balance their portfolios and choosestate contingent consumption allocations for any of the successor nodes of st,which we denote st+1 j st.

De�nition 3.2.1 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)]g

s:t:

xt+� + q�t+�zt+� = !t+� + At+�zt+��1;

for � = 0; 1; 2; :::; with z�1 = 0

some no-Ponzi scheme condition

De�nition 3.2.2 ii.P

i x�i � !i = 0 and

Pi z�i = 0:

3.2 FINANCIAL MARKETS ECONOMIES 43

3.2.1 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) (3.1)

or,

1 = Et

��u0(xt+1)

u0(xt)Rt+1

�: (3.2)

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 (3.1), 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; 0gDe�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: theprice changes with time, as information is revealed by approaching the exe-cution period T .

Example 3. The risk-free rate is know at time t and therefore, equation(3.2) applied to a 1-period bond yields

1

Rft+1= Et

��u0(xt+1)

u0(xt)

�:

Once again, note that the formula involves the conditional expectation attime t. Therefore, while the return of a risk free 1-period bond paying att+1 is known at time t, the return of a risk free 1-period bond paying at t+2is not known at time t. [.... relationship between 1 and � period bonds....from the red Sargent book]


Conditional asset pricing

Conditional versions of the beta representation hold in this economy:

Et(Rt+1)�Rft+1 = �Covt(mt+1; Rt+1)

Et(mt+1)= (3.3)

=Covt(mt+1; Rt+1)

V art(mt+1)

��V art(mt+1)

Et(mt+1)

�=: �t�t:

Recall that our basic pricing equation is a conditional expectation:

qt = Et(mt+1At+1); (3.4)

In empirical work, it is convenient to test for unconditional moment restric-tions.1 However, taking unconditional expectations of the previous equationimplies in principle a much weaker statement about asset prices than equa-tion (3.4):

E(qt) = E(mt+1At+1); (3.5)

where we have invoked the law of iterated expectations. It should be clearthat equation (3.4) implies but it is not implied by (3.5).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 testsof unconditional moment conditions.De�ne a stochastic process fitg1t=0 to be conformable if for each t, it

belongs to the time-t information set of the agent. It then follows that forany such process, 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 anadditional testable implication that only involves unconditional moments.

1Otherwise, speci�c parametric assumptions need be imposed on the stochasticprocess of the economy underlying the asset pricing equation. For instance this isthe route taken by the literature on autoregressive conditionally heteroschedastic (thatis, ARCH - and then ARCH-M, GARCH,...) models. See for instance the work by??rengle/http://pages.stern.nyu.edu/ rengle/��= 13 .


Obviously, all these implications are necessary conditions for our basic pricingequation to hold. The following result states that if we could test theseunconditional restrictions for all possible conformable processes then it wouldalso be su¢ cient. We state it without proof.

Theorem 3.2.3 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.

Predictability of 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 tofollow a random walk. (Where in turn predictability is interpreted as aproperty of e¢ cient market hypothesis, a fancy name for the asset pricingtheory exposed in these notes). Is it so? No, unless strong extra assumptionsare imposed Assume that 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,for any f"tg such that Et("t+1) = 0 at all t, we can rewrite the previousequation as

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 approximatelya martingale (a weaker notion of lack of predictability). Again under nodividends,

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

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

Frictions

[....He-Modest .... Luttmer...]


3.2.2 Bubbles

For assets whose payo¤ is made of a dividend and a capital gain, the foc�sdictate

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

TXj=1

mt;t+jdt+j

!+ limT�!1

Et

TXj=1

mt;t+jqt+j

!;

As we shall see, in�nite horizon models (with in�nitely lived agents) usu-ally satisfy the no-bubbles condition, or

limT�!1

Et

TXj=1

mt;t+jqt+j

!= 0:

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

qt = Et

1Xj=1

mt;t+jdt+j

!:

Amore general �nancial market economy is useful to study the conditionsfor the existence bubbles; we follow Santos and Woodford (1997). Let N =X1t=0S

t be the set of nodes of the tree. Recall we denoted with s0 denote theroot of the tree and with st an arbitrary node of the tree at time t. Denoteby st�1 the single (immediate) predecessor node to st. Use sT jst to indicatethat sT is some successor of st, for T > t:At each node, there are J securities traded. Let I(st) be the set of agents

which are active at node st. Let N i be the subset of nodes of the tree atwhich agent i is allowed to trade. Also, denote by N

ithe terminal nodes for

agent i.The following assumptions will not be relaxed.Assumption 1. If an agent i is alive at some non-terminal node st, she

is also alive at all the immediate successor nodes. That is,

st�N inN i=) fst+1�N : st+1jstg � N i:


Assumption 2. The economy is connected across time and states. Thisis achieved when at any state there is some agent alive and non-terminal.Formally,

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

Let q : N �! RJ be the mapping de�ning the vector of security prices ateach node st. Similarly, let d : N �! RJ denote the vector-valued mappingthat de�nes the dividends (in units of numeraire) that are paid by the assetsthat pay at node st. We shall assume that a security can pay in dividends(units of consumption) and in units of assets. Mapping b : N �! RJ

2de�nes

at each node st a J�J matrix, whose j-th column denotes the vector of assetsthat are paid at node st by asset j:Assumption 3. Assets�payo¤s, both in terms of dividends and of other

assets, are non-negative: d(st) � 0 and b(st) � 0; for any st:Each of the households alive at s0 enters the markets with an initial

endowment of securities zi!(s0): Therefore, the initial net supply of assets is

given byz!(s

0) =Xi2I(s0)

zi!(s0).

De�ne the net supply of securities at any node st, z!(st), recursively by

z!(st) = b(st)z!(s

t � 1):

We shall assume that z!(s�1) � 0: This ensures that z!(st) � 0, at all nodesstjs0; t > 0.Observe that a portfolio held at the end of st, say z(st), will generally

pay dividends (and assets) at several future states st+1; st+2, etc. We shallnow construct a mapping that will assign to each node the dividends paidby portfolio z(st). As an intermediate step, let us construct the dividendsmapping for a portfolio made of one unit of each asset that can be tradedat st. To determine the stream of dividends generated by this �canonical�portfolio at each successor node, de�ne for all srjst with r � t, the J � Jmatrix e(srjst) by

e(stjst) = IJ�Je(srjst) = b(sr)e(sr � 1jst); for all srjst and r > t:

Matrix e(srjst) is just a counter of the asset payo¤s at sr of the canonicalportfolio. Now, for all srjst with r > t, de�ne


x(srjst) = d(sr)e(sr � 1jst):This is a J-dimensional vector x(srjst) and its j-th component is the divi-dends that the j-th asset of the canonical portfolio produces at node sr, forj = 1; :::; J . Finally, the dividends mapping for the vector z(st) is simplyx(srjst)z(st); for each node srjst with r > t.De�ne an asset j to have �nite maturity if there exists a T such that

e(srjst) = 0; for all srjst and r � T > t: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) =Xi2I(st)

!i(st) � 0

at each node st. Taking into account the dividends paid by securities in unitsof good, the aggregate good supply in the economy is given by

e!(st) = !(st) + d(st)z!(st � 1) � 0:The utility function of any agent i is written

U(x) =1Xt=0

�tXst

�stui(x(st)):

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

A(st) = d(st) + q(st)b(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!(s0);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 householdspeci�c borrowing limit at each node. We assume households take the bor-rowing limits as given, just as they take security prices as given.Market Clearing conditions. At each st;X

i2I(st)

xi(st) = ew(st)Xi2I(st)

zi(st) = z!(st):

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 3.2.4 When q satis�es the no-arbitrage condition at st; there existsa set of state prices (a SDF) fm(st+1)g with m(st+1) > 0 for all st+1jst, suchthat the vector of asset prices at st can be written as

q(st) =Xst+1jst

m(st+1)A(st+1): (3.6)

Proof. As usual, proof follows from applying the separation result.Note that if for a given price process q, there are no arbitrage opportu-

nities at any st individually, then we can apply the lemma repeatedly andde�ne some state-price process m for which the pricing equation holds. LetM(st) denote the set of such processes for the subtree with root st. Onlyunder complete markets is the set M(st) a singleton.As a remark, note that completeness is an endogenous property since

one-period payo¤s A contain asset prices. Therefore, completeness cannotbe assessed ex ante but only at each given equilibrium.

De�nition 3.2.5 For any state-price process m 2M(st); de�ne the J vectorof fundamental values for the securities traded at node st by

f(st;m) =1X

T=t+1

XsT jst

m(sT )x(sT jst): (3.7)


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 3.2.6 At each st 2 N , f(st;m) is well-de�ned for any m 2M(st) and satis�es

0 � f(st;m) � q(st):

Proof. First of all, 0 � f(st;m) follows directly from non-negativity of m,the dividend, the asset payo¤, and the price processes. We therefore turn tof(st;m) � q(st): From equation (3.6), we have

q(st) =Xst+1jst

m(st+1)x(st+1jst) +Xst+1jst

m(st+1)q(st+1)e(st+1jst)

and, iterating on this equation we obtain

q(st) =

bTXT=t+1

XsT jst

m(sT )x(sT jst) +Xs bT jst

m(sbT )q(sbT )e(sbT jst)

for any bT > t: Since e(sbT jst) � 0 by construction, q(sbT ) is non-negative byde�nition of equilibrium (in the paper) and m 2 M(st) is a positive state-price vector, the second term on the right-hand-side is non-negative. So,

q(st) �bTX

T=t+1

Xst+1jst

m(sT )x(sT jst):

Note that the right-hand-side is a nondecreasing series in bT . It is boundedabove and, therefore, must converge. So,

q(st) � m(st)f(st;m)

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

�(st;m) = q(st)� f(st;m); (3.8)

for any m 2 M(st) for the J securities traded at st. It follows from theproposition that

0 � �(st;m) � q(st);


for any m 2 M(st): This corollary is known as the impossibility of negativebubbles result. Substituting (3.8) and (3.7) into (3.6) yields

�(st) =Xst+1jst

m(st+1)�(st+1)e(st+1jst):

This is known as themartingale property of bubbles: if there exists a (nonzero)price bubble on any security at date t, there must exist a bubble as well onsome securities at date T , with positive probability, at every date T > t.Furthermore, if there exists a bubble on any security at node st, then theremust have existed a bubble as well on some security at every predecessor ofthe node st.The proposition also o¤ers a corollary that deals with �nite-maturity

assets. Even with incomplete markets, we have that f j(st;m) = f j(st) forall m 2 M(st) and qj(st) = f j(st); if asset j has �nite maturiry: there areno pricing bubbles for securities with �nite maturity. Notice that wereached this conclusion just by no arbitrage.In the case of securities of in�nite maturity in an economy with incomplete

markets, the fundamental value need not be the same for all state-priceprocesses consistent with the available securities returns. But even in thiscase, we can de�ne the range of variation in the fundamental value, given therestrictions imposed by no-arbitrage.

De�nition 3.2.7 (Present value of a stream of dividends) Let x : N �! R+denote a stream of non-negative dividends. For any st, pick any m 2M(st).Then we de�ne the present value at st of x with respect to m by

Vx(st;m) =

1XT=t+1

XsT jst

m(sT )x(sT ).

Since this present value depends on the stochastic discount factor mpicked, let us now de�ne the bounds for the present value at st of dividendsx.

De�nition 3.2.8 For any st, suppose x(sr) � 0; 8srjst; r > t: De�ne

�x(st) = inf

m2M(st)fVx(st;m)g

�x(st) = sup

m2M(st)

fVx(st;m)g:


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

t) � f j(st;m) � �xj(st), for allm 2 M(st). Finally, note that �xj(st) < qj implies that there is a pricingbubble for security j.Recall that to rule out Ponzi schemes when agents are in�nitely lived, a

lower bound on individual wealth is needed. Let us de�ne a particular typeof borrowing limit.

De�nition 3.2.9 We shall say that an agent�s borrowing ability is only lim-ited by her ability to repay out of her own future endowment if

Bi(st) = �~!i(st); (3.9)

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). They are just a constraint on theasymptotic behavior of a household�s debt. They are equivalent to requir-ing that the consumption process lies in the space of measurable boundedsequences. In the case of �nitely lived agents, these borrowing limits areequivalent to imposing no-borrowing at all nonterminal nodes.An important consequence of this speci�cation is the following.

Proposition 3.2.10 Suppose that household i has borrowing limits of theform (3.9). Then the existence of a solution to the agent�s problem for givenprices q implies that �~!i(s

t) < 1; at each st 2 N i; so that there is a �niteborrowing limit at each node.

Note that if more stringent borrowing limits were imposed, i.e. Bi(st) <�~!i(s

t), we can have equilibria where �~!i(st) = 1. This is a crucial point

to understand bubbles (recall this comment in the Bewley�s example): if anagent�s endowment (that is, the whole - or part of the whole - stochasticprocess) is can be traded, then its value is on the right hand side of thepresent value budget constraint of the agent, and hence it must be �nite.But note also that even if �~!i(s

t) < 1, it is still possible that �~!(st) = 1(and, a fortiori, that �~!(st) =1) because the economy allows for a countablein�nity of agents, though we require the number of agents at each node to be�nite (recall this comment in Samuelson�s overlapping generations example).


Proposition 3.2.11 Consider an equilibrium fq; xi; zig. Suppose that the(maximum) value of aggregate wealth is �nite, i.e. �~!(st) < 1. Supposealso that 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 suchthat

�(sT )z(sT ) > KXsT jst

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

�(st)z(st) =XsT jst

m(sT )�(sT )z(sT )

whileP

sT jstm(sT )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

on the securities becomes an arbitrarily large multiple of the value of theaggregate supply of goods in the economy. The proof exploits crucially themartingale property of bubbles.It follows from this result that some agent must accumulate vast wealth.We already learned that no bubbles can arise in securities with �nite

maturity. The next theorem extends the result to securities in positive netsupply as long as we are at equilibria with �nite aggregate wealth. The proofuses the nonoptimality of the behavior implied by the previous proposition.

Theorem 3.2.12 Let fq; xi; zig be an equilibrium. Suppose that at each nodest 2 N; there exists m 2M(st) such that V~!(st;m) <1: Then

qj(sT ) = f j(sT ;m);

for all sT jst and m 2 M(st), for each security j traded at sT that has �nitematurity or positive net supply.

Note that if we have that at equilibrium �~!(st) <1; the condition of thetheorem is satis�ed.The next two corollaries to the theorem provide conditions on the primi-

tives of the model that guarantee that the value of aggregate wealth is �niteat any equilibrium.


Corollary 3.2.13 Suppose that there exists a portfolio bz�RJ+ such thatx(stjs0)bz � ~!(st); 8st 2 N:

Then the theorem holds at any equilibrium.

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

Corollary 3.2.14 Suppose that there exists an (in�nitely lived) agent andan " > 0 such that i) !i(st) � "!(st); 8st 2 N and ii) Bi(st) = �!i(s

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

The intuition for this result is as follows. The borrowing limits imposedallow all agents (and in particular the in�nitely lived one) to �issue�an IOUagainst his or her stream of endowments. Again, in equilibrium, the assetissued by the in�nitely lived agent must have a �nite price. This impliesthat a fraction of aggregate wealth has a �nite value in equilibrium, whichobviously implies that aggregate wealth is �nite.As a remark, note that these two corollaries share the same spirit. They

both state that by adding some assets to the economy, we can rule out bub-bles in equilibrium (for securities in positive net supply). Therefore, theconclusion of the paper is that bubbles are a non-robust phenom-enon (in securities in positive net supply).

Examples

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 �atmoney is in net supply and has a positive price in equilibrium, that is abubble. The following two models have equilibria with such a property.Theorem 3.3. implies that in those equilibria aggregate wealth must have anin�nite value. See the paper for the primitives of each model.1. Samuelson (1958)�s OLG model: Observe how all the assumptions of

corollary 3.5 hold except the existence of an agent that owns a fraction ofaggregate wealth.


2. Bewley (1980): Observe that now the only assumption of corollary3.5 that fails is that the borrowing constraints are more stringent than thoserequired in the corollary.3. More recent examples: Jovanovic (2007), Hong, Scheinkman, Xiong

(2006), Abreu and Brunnermeier (2003), Allen, Morris, and Shin (2003).

3.2.3 Pareto optimality

In this section we consider in�nite horizon economies with an in�nite num-ber of agents (hence double in�nity). The classic example is an overlappinggeneration economy, where at each time t = 0; 1; 2; :::1 a �nitely-lived gen-eration is born.We consider �rst the case of an Arrow-Debreu economy with a complete

set of time and state contingent markets. Assume a representative-agenteconomy with one good. Let time be indexed by t = 0; 1; 2; :::: Uncertainty iscaptured by a probability space represented by a tree with root s0 and genericnode, at time t, st 2 St. Let `1+ denote the space of non-negative boundedsequences endowed with the sup�norm.2 Let xi := fxitg1t=0 2 `1+ denote astochastic process for an agent i�s consumption, where xit : S

t �! R+ is arandom variable on the underlying probability space, for each t. Similarly,let !i = f!itg1t=0 2 `1+ be a stochastic processes describing an agent i�sendowments. Each agent i0s preferences , U i : `1+ ! R; satisfy:

U i(x) = ui0(x0) + E0[1Xt=1

�tuit(xt)]

where uit : R+ ! R satisfy the standard di¤erentiability, monotonicity, andconcavity properties, for any i > 0 and any t > 0. Obviously, 0 < � < 1denotes the discount factor.We say that an agent i is alive at st 2 N if !i (st) > 0 and uit(xt) > 0

for some xt 2 R+: We assume that !i (st) > 0 implies uit(xt) > 0 for somext 2 R+ and conversely, uit(xt) > 0 for some xt 2 R+ implies !i (st) > 0 forall st 2 St:We maintain the previous section assumptions that if an agent iis alive at some non-terminal node st, she is also alive at all the immediatesuccessor nodes, and that the economy is connected across time and states

2See Lucas-Stokey with Prescott (1989), Recersive methods in economic dynamics, Har-vard University Press, for de�nitions.


(at any state there is some agent alive and non-terminal). We �nally assumethat,

i) each agent is �nitely-lived: !i > 0 for �nitely many times t � 0;

ii) at each node st 2 St; the set of agents alive (with positive endowmentsand preferences for consumption), I( st) is �nite.

Suppose that at time zero, the agent can trade in contingent commodities.Let p := fptg1t=0 2 `1+ denote the stochastic process for prices, where pt :St �! R+, for each t. The de�nition of Arrow-Debreu equilibrium in thiseconomy is exactly as for the one with �nite agents i 2 I: Let x = (xi)i�0 :

De�nition 3.2.15 (x�; p�) is an Arrow-Debreu Equilibrium if

i. given p�;

x�i 2 argmaxfu(x0) + E0[P1

t=1 �tu(xt)]g

s:t:P1t=0 p

�t (xt � !it) = 0

ii. andP

i x�i � !i = 0:

Note that, under our assumptions,1Xt=0

p�t!it < 1; for any i � 0X

i

!i � 1:

While the de�nition of Arrow-Debreu equilibrium is unchanged once anin�nite number of agents is allowed for, its welfare properties change sub-stantially.As always, we say that x� is a Pareto optimal allocation if there does not

exist an allocation y 2 `1+ such that

U i(yi) � U i(x�i) for any i � 0 (strictly for at least one i), andIXi=1

yi � !i = 0,

Does the First welfare theorem hold in this economy? Is any Arrow-Debreuequilibrium allocation x� Pareto Optimal? Well, a quali�cation is needed.


First welfare theorem for double in�nity economies. Let (x�; p�) be anArrow-Debreu equilibrium. If at equilibrium aggregate wealth is �nite,

1Xt=0

p�t

Xi

!it

!<1;

then the equilibrium allocation x� Pareto Optimal.

The important result here is that the converse does not hold. An equi-librium with in�nite

P1t=0 p

�t (P

i !it) might not display Pareto optimal allo-

cation. In other words, with double in�nity of agents the proof of the Firstwelfare theorem breaks unless

P1t=0 p

�t (P

i !it) <1: Let�s see this.

Proof. By contradiction. Suppose there exist a y 2 `1+ which Pareto domi-nate x�:Then it must be that

1Xt=0

p�t (yit � !it) � 0; strictly for one i � 0:

Summing over i � 0, even thoughXi�0yi;Xi�0!i are �nite,

P1t=0 p

�t (P

i !it)

might not be. In this case, we cannot conclude that

1Xt=0

p�t

Xi

yit

!>

1Xt=0

p�t

Xi

!it

!:

The quali�cationP1

t=0 p�t (P

i !it) <1 is however su¢ cient to obtain

P1t=0 p

�t (P

i yit) >P1

t=0 p�t (P

i !it) and hence the contradictionX

i

yit >Xi

!it, for some t � 0:

Importantly, the proof has no implication for the converse. In otherwords, the proof is silent on

P1t=0 p

�t (P

i !it) <1 being necessary for Pareto

optimality. We shall show by example that:

-P1

t=0 p�t (P

i !it) <1 is not necessary for Pareto optimality; that is, there

exist economies which have Arrow-Debreu equilibria whose allocationsare Pareto optimal and nonetheless

P1t=0 p

�t (P

i !it) is in�nite;


- there exist economies which have Arrow-Debreu equilibria whose alloca-tions are NOT Pareto optimal.

Remark 3.2.16 The double in�nity is at the root of the possibility of in-e¢ cient equilibria in these economies. The First welfare theorem in factholds with �nite agents i 2 I even if the economy has an in�nite horizon,t � 0. This we know already. But it follows trivially (check this!) from theproof above that the First welfare theorem holds for �nite horizon economies,t = 0; 1; :::; T even if populated by an in�nite number of agents i � 0; providedof course

Pi !

i � 1:

Overlapping generation economies

We will construct in this section a simple overlapping generations economywhich displays i) Arrow-Debreu equilibria with Pareto ine¢ cient allocations(aggregate wealth is necessarily in�nite in this case, ii) Arrow-Debreu equi-libria with in�nite aggregate wealth whose allocations are Pareto e¢ cient.The economy is deterministic and is populated by two-period lived agents.

An agent�s type i � 0 indicates his birth date (all agent born at a time t aeidentical, for simplicity). Therefore, the stochastic process for the endowmentof an agent i � 0; !i, satis�es

!it > 0 for t = i; i+ 1 and = 0 otherwise.

We assume there is also an agent i = �1 with !�10 > 0. The utility functionsare as follows:

U i(xi) = u(xii) + (1� )u(xii+1); for any i � 0; 3

U�1(x�1) = u(x�10 ):

Arrow-Debreu equilibria are easily characterized for this economy.

Autarchy The economy has a unique Arrow-Debreu equilibrium (x�; p�)which satis�es:

xt�t = !tt; xt�t+1 = !

tt+1; for any t � 0;

x�1�0 = !�10

and

p0 = 1; andpt+1pt

=(1� )u0(!tt+1) u0(!tt)

; for t � 0:


The restriction p0 = 1 is the standard normalization due to the homo-geneity of the Arrow-Debreu budget constraint.

Proof. First of all,x�1�0 = !�10

follows directly from agent i = �1�s budget constraint. Then, market clearingat time t = 0 requires

x�1�0 + x0�0 = !�10 + !00:

Substituting x�1�0 = !�10 ; we obtain x0�0 = !

00:We can now proceed by induc-

tion to show that xt�t = !tt implies x

t�t+1 = !

tt+1 (using the budget constraint

of agent t), which in turn implies xt+1�t+1 = !t+1t+1 (using the market clearingcondition at time t+ 1:The characterization of equilibrium prices then follows trivially from the

�rst order conditions of each agent�s maximization problems.It is convenient to specialize this economy to a simple stationary example

where

!�10 = �; !tt = 1� �; !tt+1 = �; for any t � 0;u(x) = lnx:

Then, at the Arrow-Debreu equilibrium,

x�1�0 = �; xt�t = 1� �; xt�t+1 = �; for any t � 0;

p�t =

�

1� 1� ��

�t�1:

A symmetric Pareto optimal allocation is as follows (it is straightforwardto derive, once symmetry is imposed): at the Arrow-Debreu equilibrium,

x�10 = ; xt�t = 1� ; xt�t+1 = ; for any t � 0:

It follows then that,

The Arrow-Debreu equilibrium (autarchy) is Pareto e¢ cient if and only if

� �:


Proof. The case = � is obvious. If < � all agents i � 0 preferthe allocation

�xii = 1� ; xii+1 =

�to autarchy, but agent i = �1 prefers

x�1�0 = � to x�10 = . Autarchy is then Pareto e¢ cient. If otherwise > �all agents i � 0 prefer the allocation

�xii = 1� ; xii+1 =

�to autarchy, and

agent i = �1 as well prefers x�10 = to x�1�0 = �. Autarchy is then NOTPareto e¢ cient.Furthermore, note that in this economy, the aggregate endowment

Pi�0 !

it =

1; for any t � 0; and hence the value of aggregate wealth is:

1Xt=0

p�t

Xi

!it

!=

1Xt=0

p�t =

1Xt=0

�

1� 1� ��

�t�1:

It follows that

The value of aggregate wealth is �nite when

< a

and autarchy is e¢ cient. On the other hand, the value of aggregatewealth is in�nite when

� a:In this case, autarchy is e¢ cient if = a and ine¢ cient when > a:

Note that we have in fact shown what we were set to from the beginningof this section: i) Arrow-Debreu equilibria with Pareto ine¢ cient allocations(aggregate wealth is necessarily in�nite in this case, ii) Arrow-Debreu equi-libria with in�nite aggregate wealth whose allocations are Pareto e¢ cient.

Bubbles in OLG economies We have seen previously that, when thevalue of the aggregate endowment is in�nite, bubbles in in�nitely-lived pos-itive net supply assets might arise. We shall now show that this is the casein the overlapping generation economy we have just studied. Interestingly, itwill turn out that bubbles, in this economy might restore Pareto e¢ ciency.(This is special to overlapping generations economies, by no means a generalresult).Suppose agent i = �1 is endowed with a in�nitely-lived asset, in amount

m: The asset pays no dividend ever: it is then interpreted to be �at money.Anypositive price for money, therefore, must be due to a bubble. Let the price of


money, in units of the consumption good at time t = 0; at time t; be denotedqmt : We continue to normalize p0 = 1: The Arrow-Debreu budget constraintsin this economy for agent i = t � 0 can be written as follows:

1Xt=0

pt(xt � !it) = pt(xtt � !tt) + pt+1(xtt+1 � !tt+1) = 0;

or, denoting st the demand for money of agent i = t � 0:

ptxtt + q

mt s

t = !ttpt+1(x

tt+1 � !tt+1) = qmt+1s

t:

The budget constraint for agent i = �1 is:

x�10 = !00 + qm0 m:

Turning back to the stationary economy example where

!�10 = �; !tt = 1� �; !tt+1 = �; for any t � 0;u(x) = lnx;

we can easily characterize equilibria. Suppose in particular that > �and autarchy is ine¢ cient. We restrict the analysis to following stationarityrestriction:

qmt = qm for any t � 0:

The autarchy allocation is obtained as an Arrow-Debreu equilibrium of thiseconomy, for

qm� = 0 andp�t+1p�t

=

1� 1� ��

:

The Pareto optimal allocation

x�10 = ; xt�t = 1� ; xt�t+1 = ; for any t � 0

is also obtained an Arrow-Debreu equilibrium of this economy, for

qm� = � �m

andp�t+1p�t

= 1:


It is straightforward to check that these are actually Arrow-Debreu equi-libria of the economy. Note that Pareto optimality is obtained at equilibriumfor qm� > 0; that is, when money has positive value and hence a bubble exist.At this equilibrium, prices p�t are constant over time and hence the value ofthe aggregate wealth is in�nite.

Remark 3.2.17 The stationary overlapping generation example introducedin this section has been studied by Samuelson (1958). The fundamental intu-ition for the role of money in this economy is straightforward: money allowsany agent i � 0 to save by acquiring money (in exchange for goods) at timet = i from agent i � 1 and then transfering the same amount of money (inexchange for the same amount of goods) to agent i + 1 at time t = i + 1:Money, in other words, serves the purpose of a pay-as-you-go social securitysystem; and in fact such a system, implemented by an in�nitely lived agent(like a benevolent government), could substitute for money in this economy.(Try and set it up formally!) Furthermore, note that, for money to havevalue in this economy, we need > �. Consistently with the interpretationof money as a social security mechanism, the condition > � is interpretedto require that i) the endowment of any agent i � 0 at time t = i + 1, �;be relatively small and that ii) any agent i � 0 discounts relatively little thefuture, that is, � =

1� is high.

Remark 3.2.18 Large-square economies, with a continuum of agents andcommodities, are studied by Kehoe-Levine-Mas Colell-Woodford (1991), "Gross-substitutability in large-square economies," Journal of Economic Theory, 54(1),1-25.

3.3 Bewley economies

In this section we consider two economies with incomplete markets and idio-synchratic shocks which have been studied in macroeconomics. The �rst,economies with earning risk, are economies characterized by:

agents face stationary idiosynchratic endowment shocks (with/out an ag-gregate component), but

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

3.3 BEWLEY ECONOMIES 63

The second class of economies, referred to as economies with investmentrisk, are characterized 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 economies can be somewhat extended to allow for production. Wediscuss these extension in a future chapter.

3.3.1 Earning risk

The prototypical earning risk economy is an economy with idiosynchraticshocks in which asset trading is restricted to a bond, which trades at time tfor a price qt normalized 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 savingsis an exogenous sequence rt; see Aiyagari (1994) and Ljungvist and Sargent(2004), ch. 17. An equilibrium is still characterized by a policy function ofthe form

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

and a distribution �t(zt; st); de�ned recursively from an initial given distrib-ution �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):

A straightforward modi�cation/reinterpretation/extension of this econ-omy, has production to endogeneize the rate of return (on capital; re-interpret

4We drop the dependence of the policy function from the whole sequence rt; for nota-tional simplicity.


wealth as capital) and a wage rate. Let F (Kt; Nt) the aggregate productionfunction at time t, in terms of aggregate capital Kt and labor Nt: Assumee.g., that labor supply is �xed and !t(s) = wts, where wt is the wage rate attime t, and the shock 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. Typicallywe assume write it as

atF (Kt; Nt):

3.3.2 Investment risk

Suppose instead each agent faces an idiosynchratic exogenous rate of returnon savings, a mapping r : f1; 2; :::; Sg �! R+. We maintain the assumptionthat each agent can only save (not borrow) at the rate r: zt � 0 for anyt � 0. Given the process r;5 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:

5Formally, the realization for s�1 is also given. Assume also agents are endowed withno portfolio positions on bonds at time 0: zi(s�1) = 0; for all i 2 I:

3.3 BEWLEY ECONOMIES 65

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

�t+1(zt+1) =Xst+12S

�t+1(zt+1; st+1)

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

�(z) = limt!1

�t(z):

3.3.3 Limit incomplete market economies

Let�s go back to economies with no trading restrictions and in�nitely livedagents. Of course, we still need to impose some borrowing limit to rule outPonzi schemes but we know how to do that in a non-binding way, as welearned from Santos and Woodford (1997).In a series of papers, Telmer (1993), Aiyagari (1994) and Krusell and

Smith (1998) among others, di¤erent authors have found support for a puz-zling result. Even though theoretically the e¤ects of complete or incompletemarkets on equilibrium allocations and prices are crucial, empirically, they itdoes not seem to matter signi�cantly. These are all in�nite-horizon economieswith agents facing idiosyncratic risk due to stochastic endowments and a se-riously incomplete asset structure.The next two papers illuminate the question from the theoretical view-

point. The current state of the literature is that the stationarity or not ofthe individual endowment process is key to the e¤ects of incomplete markets.This is the result from Levine and Zame (2002). A second contribution isConstantinides and Du¢ e (1996), who show that with incomplete marketsand nonstationary endowment processes (i.e. permanent shocks), �anythingis possible�.For short horizons, we know that market incompleteness generally matters

because of agents�inability to insure against bad shocks. However, for longhorizons, market incompleteness may not matter if traders can self insure i.e.if they can borrow in bad times and save in good times.Consider the same economy we studied for bubbles, but let�s specialize

it to our purposes. Let uncertainty de�ne a Markov process on the treeN = X1

t=0St as follows:

each node st has S successors, that is, st+1jst = f1; 2; :::; Sg ;


prob( st+1jst = s : st = s0 ) = prob (sjs0), for any s; s0 2 f1; 2; :::; Sg;obviously prob (sjs0) de�nes a transition probability of a Markov chainwith state space f1; 2; :::; Sg.

We assume the Markov chain is recurrent (e.g., it is su¢ cient thatprob (sjs0) > 0, for any s; s0 2 f1; 2; :::; Sg)At each node st, J short-lived, actually one-period, securities are traded.

Let the dividend process be the map d : f1; 2; :::; Sg �! R+ ; a J-dimensional Markov process in our probability space. Assume also that ateach node st, a riskless bond, yielding one unit of consumption at each st+1jst,is available for trade. All agents are alive and trade at all periods t > 0.Furthermore, each agent�s i endowment process is a mapping !i : f1; 2; :::; Sg �!

R+ , once again a Markov process in our probability space.De�nition of equilibrium in this economy are just specialized versions

of the de�nitions in the economy we studied for bubbles (in particular, weimpose the same No Ponzi scheme condition).Let�s assume the following:

the economy has no aggregate risk, that is,

! =Xi

!i(s) is independent of s;

preferences satisfy precautionary savings, that is,

Dui(x) is (weakly) convex:

Let !i denote the long-run average endowment (permanent income) foragent i: It is well de�ned because under our assumptions, the Markov processfor endowment has a stationary distribution. In fact, let such stationarydistribution associate probability prob�(s) to any s 2 f1; 2; :::; Sg ; then

!i =Xs

prob�(s)!i(s)

It is easy to show that, the Pareto e¢ cient allocations of this economy aregiven by the I-tuples of �xed shares of the constant long-run average endow-ment ! =

Pi !

i. In particular, the complete market equilibrium allocation(hence Pareto e¢ cient) for this economy is characterized by each agent iconsuming his/her long-run average endowment (permanent income) at eachnode st:


Theorem 3.3.1 For this economy there exist a discount factor � su¢ cientlyclose to 1, such that every equilibrium allocation process xi is "close" toperfect risk sharing,

xi(st) � !i

(in the sense that the time-discounted probability that equilibrium consump-tions deviate from the perfect risk sharing allocation by more than a givenamount is small).

The proof provides a lower bound on equilibrium utility by constructinga budget feasible plan whose utility is almost that of constant average con-sumption. A crucial step in the argument is establishing that the risklessinterest rate is bounded above, with a bound close to zero. This is importantbecause the budget feasible plan they construct is �nanced by borrowing,and a low interest rate makes borrowing easy.In general, in the presence of aggregate risk, market incompleteness mat-

ters even if endowment processes are stationary (i.e. shocks are transitory).The reason is the following. When there is aggregate risk, the upper boundon the interest rate need not obtain; when the aggregate endowment is low,many traders will want to borrow, and this demand for loans may drive upthe riskless interest rate. A high interest rate interferes with risk sharingbecause it makes borrowing di¢ cult. Summing up, aggregate risk mattersbecause it a¤ects asset prices.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,

patient agents, market incompleteness will not matter if shocks are transitoryand risk is purely idiosyncratic. When there is aggregate uncertainty or morethan one consumption good, market incompleteness matters, in general.The next paper adds to the question by showing that if shocks are per-

manent, market incompleteness matters too.

3.4 Asymmetric information economies

[...survey...]


3.4.1 Lack of Commitment Economies

Consider the following economy, populated by a �nite set of in�nitely-livedagents, i 2 I: Let N = X1

t=0St be the set of nodes of the tree, s0 the root of

the tree, st an arbitrary node of the tree at time t. Use st+� jst to indicatethat st+� is some successor of st, for � > 0:At each node, there are S securities in zero-net supply traded with lin-

early independent payo¤s: �nancial markets are complete. Without loss ofgenerality we let the �nancial assets be a full set of Arrow securities: A = IS(the S�dimensional identity matrix).Let q : N �! RS be the mapping de�ning the vector of asset prices at

each node st.At each node st, each households has an endowment of numeraire good

of !i(st) > 0; aggregate endowment is then

!(st) =Xi2I!i(st) � 0

at each node st. At the root of the tree the expected utility of agent i 2 I is:

U i(x; s0) = ui(x(s0) +1Xt=1

�tXstjs0

prob(st��s0 ) ui(x(st ��s0 )):

Any agent i 2 I; at any node st 2 N can default (more precisely: cannotcommit not to default). If he does default at a node st, he is forever kept outof �nancial markets and is therefore limited to consume his own endowment!i(st+� jst ) at any successor node st+� jst 2 N:An agent i 2 I; therefore, will default at a node st on allocation xi if

U i(xi; st) < U i(!i; st):

The notion of equilibrium we adopt for this economy will be the one usedby Prescott and Townsend for economies with asymmetric information wherea no-default constraint

U i(xi; st) � U i(!i; st); for any st 2 N;

takes the place of the incentive compatibility constraint. In particular weshall choose the notion of equilibrium introduced in the remark, where these


constraints are interpreted as constraints on the set of tradable allocation(rather than rationality restrictions on price conjectures).Let ((x�i)i ; p

�) be an Arrow-Debreu Equilibrium if x�i; for any agenti 2 I; solves

maxxiU i(xi; s0)

s.t.

p�(xi � !i) = 0; and

U i(xi; st) � U i(!i; st); for any st 2 N

and markets clear: Xi

x�i � !i = 0:

This problem is formulated and studied by T. Kehoe and D. Levine,Review of Economic Studies, 1993. Note that

i) the value of aggregate wealth for each agent i 2 I is necessarily �nite atequilibrium:

p�!i <1

(and I is �nite by assumption)

ii) the set of constraints which guarantee no-default at equilibrium do notdepend on equilibrium prices

U i(xi�; st) � U i(!i; st); for any st 2 N:

Let now incentive constrained Pareto optimal allocations be those whichsolve

max(xi)i2I

Xi2I�iU i(xi; s0)

s.t. Xi2Ixi � !i = 0; and

U i(xi�; st) � U i(!i; st); for any st 2 N;

for some � 2 RI+ such thatP

i2I �i = 1:


It follows easily then that

Any Arrow-Debreu Equilibrium allocation of this economy is constraintPareto optimal.6

Consider now a �nancial market equilibrium for this economy. The ob-jective is to capture the no-default constraints by means of appropriate bor-rowing constraints. In other words we want to set borrowing constraints asloose as possible provided no agent would ever default. To this end we needto write exploit the recursive structure of the economy. Let �i(st+1 jst ) de-note the portfolio of Arrow security paying o¤ in node st+1 acquired at thepredecessor node st: The borrowing constraints an agent i 2 I will face atnode st will then take the form

�i(st+1��st ) � Bi(st+1 ��st ); for any st+1 ��st ;

and Bi(st+1 jst ) must be chosen to be as loose as possible provided it inducesagent i 2 I not to default at any node st+1 jst : Formally, the value functionof the problem of agent i 2 I at any node st 2 N when i) the agent entersstate st 2 N with �i units of the Arrow security which pays at st 2 N; andthe agent faces borrowing limits Bi(st+1 jst ), can be constructed as follows:

V i(�i; st) = maxxi;�i(st+1jst ) ui(xi) + �

Pst+1jst prob(s

t+1 jst )V i(�i(st+1 jst ); st)

s:t:

xi +P

st+1jst �i(st+1 jst )q(st+1 jst ) = �i + !i(st+1 jst ); and

�i(st+1 jst ) � Bi(st+1 jst ):The condition that the borrowing limitsBi(st+1 jst ) be as loose as possible

is then endogenously determined as

V i(Bi(st+1��st ; st) = U i(!i; st+1).

It can be shown that V i(Bi(st+1 jst ; st) is monotonic (decreasing) in its �rstargument, for any st 2 N: Borrowing limits Bi(st+1 jst ) are then uniquelydetermined at any node. Note that they are determined at equilibrium,however, as they depend on the value function V i(:; st):

6In fact, in the Kehoe and Levine paper, L > 1 commodities are traded at each nodeand the conquence of default is that trade is restricted to spot markets at any future node.In this economy the non-default constraints depend on spot prices and Arrow-Debreuequilibrium allocations are not incentive constrained optimal.


Market clearing brings no surprises:Xi

xi � !i = 0:

We can now show the following.The autarchy allocation

x�i(st) = !i(st); for any st 2 N;

can always be supported as a �nancial market equilibrium allocation withborrowing constraints

Bi(st+1��st ) = 0; for any st 2 N:

Note in particular that V i(Bi(st+1 jst ; st) = U i(!i; st+1) is satis�ed at thisequilibrium.Let

q(st+2jst) = q(st+2jst+1)q(st+1jst); for t = 0; 1; :::We can then show the following.Any �nancial market equilibrium allocation (x�i)i2I whose supporting prices

q�(st+1 jst ) satisfy

!i�s0�+Xt�0

Xst+1js0

q�(st+1��s0 )!i(st+1 ��s0 ) <1

is an Arrow-Debreu equilibrium allocation supported by prices

p�(s0) = 1; p�(st) = q�(st��s0 ):

This can be proved by repeatedly solving forward the �nancial marketequilibrium budget constraints, using the relation between Arrow-Debreuand �nancail market equilibrium prices in the statement. The solution isnecessarily the Arrow-Debreu budget constraint only if limT!1 q

�(sT js0 ) =0, which is the case if the value of aggregate wealth is �nite at the �nancialmarket equilibrium.In this case, then the �nancial market equilibrium allocation (x�i)i2I is

constrained Pareto optimal. When (x�i)i2I is not autarchic (easy to show byexample that robustly, such �nancial market equilibrium allocations exist)it Pareto dominates autarchy (as autarchy is always budget feasible and


satis�es the no-default constraint). In this case the autarchy allocation is notconstrained Pareto optimal and the value of some agent�s wealth is in�niteat the autarchy equilibrium.7

Remark 3.4.1 The autarchy allocation asset prices must satisfy

q�(st+1��st ) � max

i2I

ui0(!i(st+1))

ui0(!i(st)):

7Bloise and Reichlin (2009) prove that in this economy �nancial market equilibria within�nite value of aggregate wealth other than autrachy exist. Some of these equilibriaare e¢ cient and some are not. The machinery used to prove these results is related tothe classical analysis of Overlapping Generations and Bewley models. Not surprisingly,bubbles also arise.

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