Alberto Bisin

Lecture Notes on Financial Economics:

Two-Period Exchange Economies

September, 20091

1 Dynamic Exchange Economies

In a two-period pure exchange economy we study �nancial market equilibria. Inparticular, we study the welfare properties of equilibria and their implicationsin terms of asset pricing.In this context, as a foundation for macroeconomics and �nancial economics,

we study

su¢ cient conditions for aggregation, so that the standard analysis of one-goodeconomies is without loss of generality,

su¢ cient conditions for the representative agent theorem, so that the standardanalysis of single agent economies is without loss of generality

The No-arbitrage theorem and the Arrow theorem on the decentralizationof equilibria of state and time contingent good economies via �nancial marketsare introduced as useful means to characterize �nancial market equilibria.

1.1 Time and state contingent commodities

Consider an economy extending for 2 periods, t = 0; 1. Let i 2 f1; :::; Ig denoteagents and l 2 f1; :::; Lg physical goods of the economy. In addition, the stateof the world at time t = 1 is uncertain. Let f1; :::; Sg denote the state spaceof the economy at t = 1. For notational convenience we typically identify t = 0with 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+; for

any s = 0; :::S. Let ui : X �! R denote agent i�s utility function. We willassume:

Assumption 1 !i 2 Rn++ for all i1Thanks to Francesc Ortega for research assistance with the �rst draft of these notes - so

long ago I do not with to remember.

1

Assumption 2 ui is continuous, strongly monotonic, strictly quasiconcave andsmooth, for all i (see Magill-Quinzii, p.50 for de�nitions and details). Further-more, 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. Thatis, contracts for the delivery of goods at time 1 contingently to the realizationof 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 0 thedelivery 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 as2

�0(xi0 � !i0) +

SXs=0

�s(xis � !is) = 0 (1)

De�nition 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�i � !is = 0, for any s = 0; 1; :::; S

Observe that the dynamic and uncertain nature of the economy (consump-tion occurs at di¤erent times t = 0; 1 and states s 2 S) does not manifests itselfin the analysis: a consumption good l at a time t and state s is treated simply asa di¤erent commodity than the same consumption good l at a di¤erent time t0

or at the same time t but di¤erent state s0. This is the simple trick introducedin Debreu�s last chapter of the Theory of Value. It has the fundamental im-plication that the standard theory and results of static equilibrium economiescan be applied without change to our dynamic) environment. In particular,then, under the standard set of assumptions on preferences and endowments,an equilibrium exists and the First and Second Welfare Theorems hold.3

2We write the budget constraint with equality. This is without loss of generality undermonotonicity of preferences, an assumption we shall maintain.

3Having set de�nitions for 2-periods Arrow-Debreu economies, it should be apparent how ageneralization to any �nite T -periods economies is in fact e¤ectively straightforward. In�nitehorizon will be dealt with in successive notes.

2

De�nition 2 Let (x�; ��) be an Arrow-Debreu equilibrium. We say that x� isa Pareto optimal allocation if there does not exist an allocation y 2 XI suchthat

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

2:IXi=1

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

Theorem 3 Any Arrow-Debreu equilibrium allocation x� is Pareto Optimal.

Proof. By contradiction. Suppose there exist a y 2 X such that 1) and 2)in the de�nition of Pareto optimal allocation are satis�ed. Then, by 1) in thede�nition of Arrow-Debreu equilibrium, it must be that

��0(yi0 � !i0) +

SXs=0

��s(yis � !is) � 0;

for all i; and

��0(yi0 � !i0) +

SXs=0

��s(yis � !is) > 0;

for at least one i.Summing over i, then

��0

IXi=1

(yi0 � !i0) +SXs=0

��s

IXi=1

(yis � !is) > 0

which contradicts requirement 2) in the de�nition of Pareto optimal allocation.

The proof exploits strict monotonicity of preferences. Where?

1.2 Financial market economy

Consider the 2-period economy just introduced. Suppose now contingent com-modities are not traded. Instead, agents can trade in spot markets and inj 2 f1; :::; Jg assets. An asset j is a promise to pay ajs � 0 units of good l = 1in state s = 1; :::; S.4 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 =

0@ a11 ::: aJ1::: :::a1S ::: aJS

1A :It will be convenient to de�ne as to be the s-th row of the matrix. Note that itcontains the payo¤ of each of the assets in state s.

4The non-negativity restriction on asset payo¤s is just for notational simplicity.

3

Let p = (p0; p1; :::; pS), where ps 2 RL+ for each s, denote the spot price vectorfor goods. That is, for a price pls agents trade one unit of good l in state s:Recall the de�nition of prices for state contingent commodities in Arrow-Debreueconomies, denoted �: Note the di¤erence. Let good l = 1 at each date and staterepresent the numeraire; that is, p1s = 1, for all s = 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.5 Note that the prices ofassets 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 consumption

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:

De�nition 4 A Financial markets equilibrium is a (x�; z�; p�; q�) such that

1: x�i 2 argmaxui(xi) s.t.

p0(xi0 � !i0) = �qzi; and

ps(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 care about.This is because i) Arrow-Debreu markets are perhaps too demanding a require-ment, and especially because ii) we are interested in �nancial markets and assetprices q in particular. Arrow-Debreu equilibrium will be a useful concept insofaras it represents a benchmark (about which we have a wealth of available results)against which to measure Financial markets equilibrium.

De�nition 5 Remark 6 The economy just introduced is characterized by assetmarkets in zero net supply, that is, no endowments of assets are allowed for. Itis 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: the projectionof 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.

5Quantities will be row vectors and prices will be column vectors, to avoid the annoyinguse of transposes.

4

1.3 No Arbitrage

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

W =

��qA

�:

De�nition 7 W satis�es the No-arbitrage condition if there does not exist a

there does not exist a z 2 RJ such that Wz > 0:6

The No-Arbitrage condition can be equivalently formulated in the followingway. 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 (with assetstructure A). By strong monotonicity of preferences, No-arbitrage is equivalentto requiring the agent�s problem to be well de�ned. The next result is remark-able since it provides a foundation for asset pricing based only on No-arbitrage.

Theorem 8 (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 provides apricing formula for assets:

�̂W =

0@ :::

��̂0qj + �̂1aj1 + :::+ �̂SajS

:::

1A =

0@ :::0:::

1AJx1

and, rearranging, we obtain for each asset j,

qj = �1aj1 + :::+ �Sa

jS ; for �s =

�̂s�̂0

(2)

6Wz > 0 requires that all components of Wz are � 0 and at least one of them > 0:

5

Note how the positivity of all components of �̂ was necessary to obtain (2).Proof. =) De�ne the simplex in RS+1+ as � = f� 2 RS+1+ :

PSs=0 � s = 1g.

Note that by the No-arbitrage condition, < W >T� is empty. The proof hinges

crucially on the following separating result, a version of Farkas Lemma, whichwe shall take without proof.

Lemma 9 Let X be a �nite dimensional vector space. Let K be a non-empty,compact and convex subset of X. Let M be a non-empty, closed and convexsubset of X. Furthermore, let K and M be disjoint. Then, there exists �̂ 2Xnf0g such that

sup�2M

�̂� < inf�2K

�̂� :

Let X = RS+1+ , K = � and M =< W >. Observe that all the requiredproperties hold and so the Lemma applies. As a result, there exists �̂ 2 Xnf0gsuch that

sup�2<W>

�̂� < inf�2�

�̂� : (3)

It remains to show that �̂ 2 RS+1++ : Suppose, on the contrary, that there issome s for which �̂s � 0. Then note that in (3 ), the RHS� 0: By (3), then,LHS < 0: But this contradicts the fact that 0 2< W > .We still have to show that �̂W = 0, or in other words, that �̂� = 0 for all

� 2< W >. Suppose, on the contrary that there exists � 2< W > such that�̂� 6= 0: Since < W > is a subspace, there exists � 2 R such that �� 2< W >and �̂�� is as large as we want. However, RHS is bounded above, which impliesa contradiction.(= The existence of �̂ 2 RS+1++ such that �̂W = 0 implies that �̂� = 0

for all � 2< W > : By contradiction, suppose 9�� 2< W > and such that�� 2 RS+1+ nf0g: Since �̂ is strictly positive, �̂�� > 0; the desired contradiction.

A few �nal remarks to this section.

Remark 10 An asset which pays one unit of numeraire in state s and nothingin all other states (Arrow security), has price �s according to (2). Such assetis called Arrow security.

Remark 11 Is the vector �̂ obtained by the No-arbitrage theorem unique? No-tice how (??) de�nes a system of J equations and S unknowns, represented by�. 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). In general,then R(q) will have dimension S � J 0. It follows then that, in this case, theNo-arbitrage theorem restricts �̂ to lie in a S�J 0+1 dimensional set. If we hadS linearly independent assets, the solution set has dimension zero, and there isa unique � vector that solves (??). The case of S linearly independent assets isreferred to as Complete markets.

6

Remark 12 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 =�s

probs. Then

qj = E (maj)

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

1.4 Equilibrium economies and the stochastic discount fac-tor

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 and studyasset prices in a full-�edged economy. Among other things, this will allow us toprovide some economic content to the vector �Recall the de�nition of Financial market equilibrium. LetMRSis(x

i) denoteagent i�s marginal rate of substitution between consumption of the numerairegood 1 in state s and consumption of the numeraire good 1 at date 0:

MRSis(xi) =

@ui(xis)

@xi1s@ui(xi0)

@xi10

Let MRSi(xi) =�: : :MRSis(x

i) : : :�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 First Order Conditions (necessary and su¢ cient for a maxi-

mum under the assumption of strict quasi-concavity of preferences) with respectto zij of the individual problem for an arbitrary price vector q, we obtain that

qj =SXs=1

probsMRSis(x

i)ajs = E�MRSi(xi) � aj

�; (4)

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 one moreunit of asset j, qj , is equalized to the marginal valuation of that agent for theasset�s payo¤,

PSs=1 probsMRS

is(x

i)ajs.Compare equation (4) to the previous equation (2). Clearly, at any equilib-

rium, condition (4) has to hold for each agent i. Therefore, in equilibrium, thevector of marginal rates of substitution of any arbitrary agent i can be used to

7

price assets; that is any of the agents�vector of marginal rates of substitution(normalized by probabilities) is a viable stochastic discount factor m:

In other words, any vector�: : :

MRSis(xi)

probs: : :�belongs to R(q) and is hence a

viable � for the asset pricing equation (2). But recall that R(q) is of dimensionS�J 0; where J 0 is the e¤ective dimension of the asset space. The higher the thee¤ective dimension of the asset space (sloppily said, the larger �nancial mar-kets) the more aligned are agents�marginal rates of substitution at equilibrium(sloppily said, the smaller are unexploited gains from trade at equilibrium). Inthe extreme case, when markets are complete (that is, when the rank of A isS), the set R(q) is in fact a singleton and hence the MRSi(xi) are equalizedacross agents i at equilibrium: MRSi(xi) =MRS; for any i = 1; :::; I:

Problem 13 Write the Pareto problem for the economy and show that, at anyPareto optimal allocation, x; it is the case that MRSi(xi) = MRS; for anyi = 1; :::; I: Furthermore, show that an allocation x which satis�es the feasibilityconditions (market clearing) for goods and is such that MRSi(xi) =MRS; forany i = 1; :::; I: is Pareto optimal.

We conclude that, when markets are Complete, equilibrium allocations arePareto optimal. That is, the First Welfare theorem holds for Financial marketequilibria when markets are Complete.

Problem 14 (Economies with bid-ask spreads) Extend our economy by assum-ing that, given an exogenous vector 2 RJ++:

the buying price of asset j is qj + j

whilethe selling price of asset j is qj

for any j = 1; :::; J:and exogenous. Write the budget constraint and the FirstOrder Conditions for an agent i�s problem. Derive a generalized asset pricingrelation (not an equation, is it?) that relates MRSi(xi) to asset prices.

1.5 Arrow theorem

The Arrow theorem is the fondamental decentralization result in �nancial eco-nomics. It states su¢ cient conditions for a form of equivalence between theArrow-Debreu and the Financial market equilibrium concepts. It was essen-tially introduced by Arrow (1952). The proof of the theorem introduces a re-formulation of the budget constraints of the Financial market economy whichfocuses on feasible wealth transfers across states directly, on the span of 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.

8

Proposition 15 Let (x�; ��) represent an Arrow-Debreu equilibrium. Supposerank(A) = S (�nancial markets are Complete). Then (x�; z�; p�; q�) is a Fi-nancial market equilibrium, where

�� = p��: : :MRSis(x

i�)

probs: : :

�and

q� =SXs=1

probsMRSis(x

i�)as

Proof. Financial market equilibrium prices of assets q� satisfy No-arbitrage.There exists then a vector �̂ 2 RS+1++ such that �̂W = 0; or q� = �A. The budgetconstraints 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 constraintsat time 1 in vector form, we obtain:

p�0�xi�0 � !i0

�+

SXs=1

�sp�s

�xi�s � !is

�= 0 (5)266664

::

p�s�xi�s � !is

�::

377775 2 < A > (6)

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

266664::

p�s�xi�s � !is

�::

377775 2<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:

Furthermore, by No-arbitrage

q� =SXs=1

probsMRSis(x

i�)as:

Finally, using �s =MRSis(x

i�)probs

, for any s = 1; :::; S; proves the result. (Recallthat, with Complete markets MRSi(xi�) =MRS; for any i = 1; :::; I.)

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2 Constrained Pareto Optimality

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

Proposition 16 Let (x�; z�; p�; q�) be a Financial market equilibrium of aneconomy 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 17 At a Financial Market Equilibrium (x�; z�; p�; q�) of an incom-plete �nancial market economy, that is, of an economy with rank(A) < S, theallocation x� is generically7 not Pareto Optimal.

Proof. From the proof of Arrow theorem, we can write the budget con-straints of the Financial market equilibrium as:

p�0�xi�0 � !i0

�+

SXs=1

�sp�s

�xi�s � !is

�= 0 (7)266664

::

p�s�xi�s � !is

�::

377775 2 < A > (8)

for some � 2 RS++: Pareto optimality of x�requires that there does not exist anallocation y such that

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

2:IXi=1

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

Reproducing the proof of the First Welfare theorem, it is clear that, if such

a y exists, it must be that

266664::

p�s�yi�s � !is

�::

377775 =2< A >; for some i = 1; :::; I;

7We say that a statement holds generically when it holds for a full Lebesgue-measure subsetof the parameter set which characterizes the economy. In these notes we shall assume that thean economy is parametrized by the endowments for each agent, the asset payo¤ matrix, anda two-parameter parametrization of utility functions for each agent; see Magill-Shafer, ch. 30in W. Hildenbrand and H. Sonnenschein (eds.), Handbook of Mathematical Economics, Vol.IV, Elsevier, 1991.

10

otherwise the allocation y would be budget feasible for all agent i at the equi-librium prices. Generic Pareto sub-optimality of x� follows then directly fromthe following Lemma, which we leave without proof.8

Lemma 18 Let (x�; z�; p�; q�) represent a Financial Market Equilibrium of aneconomy with rank(A) < S: For a generic set of economies, the constraints266664

::

p�s�xi�s � !is

�::

377775 2< A > are binding for some i = 1; :::; I.

Remark 19 The Lemma implies a slightly stronger result than generic Paretosub-optimality of Financial market equilibrium for economies with incompletemarkets. It implies in fact that a Pareto improving allocation can be foundlocally around the equilibrium, as a perturbation of the equilibrium.

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�nea weaker notion of social welfare, Constrained Pareto optimality, by restrict-ing the 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 restricted tolie in the span of the payo¤ matrix. That can be interpreted as the economy�s��nancial technology�and it seems reasonable to impose the same technologi-cal restrictions on the planner�s reallocations. The formalization of an e¢ ciencynotion capturing this idea follows. Let xit=1 =

�xis�Ss=1

2 RSL+ ; and similarly

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

De�nition 20 (Diamond, 1968; Geanakoplos-Polemarchakis, 1986) Let (x�; z�; p�; q�)represent a Financial market equilibrium of an economy whose consumption setat time t = 1 is restricted by

xit=1;2 B(pt=1); for any i = 1; :::; I8The proof can be found in Magill-Shafer, ch. 30 in W. Hildenbrand and H. Sonnenschein

(eds.), Handbook of Mathematical Economics, Vol. IV, Elsevier, 1991. It requires mathemat-ical tecniques from di¤erential topology which are not appropriate to be introduced in thiscourse.

11

In 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 = 1 openedafter each agent i = 1; :::; I has received income transfer A�i:

The constraint on the consumption set restricts only time 1 consumptionallocations. More general constraints are possible but these formulation is con-sistent with the typical frictions we encounter in economics, e.g., on �nancialmarkets. It is important that the constraint on the consumption set dependsin general on g�t=1(!; �), that is on equilibrium prices for spot markets openedat t = 1 after income transfers to agents. It implicit identi�es income trans-fers (besides consumption allocations at time t = 0) as the instrument availablefor Constrained Pareto optimality; that is, it implicitly constrains the plannerimplementing Constraint Pareto optimal allocations to interact with markets,speci�cally to open spot markets after transfers. On the other hand, the plan-ner is able to anticipate the spot price equilibrium map, g�t=1(!; �); that is, tointernalize the e¤ects of di¤erent transfers on spot prices at equilibrium.

Proposition 21 Let (x�; z�; p�; q�) represent a Financial market equilibrium ofan economy with complete markets (rank(A) = S) and whose consumption setat time t = 1 is restricted by

xit=1 2 B � RSL+ ; for any i = 1; :::; I

In this economy, the allocation x� is Constrained Pareto optimal.

Crucially, markets are Complete and B is independent of prices. The proofis then a straightforward extension of the First Welfare theorem combined withArrow theorem. Constraint Pareto optimality of Financial market equilibriumallocations is guaranteed as long as the constraint set B is exogenous.

Proposition 22 Let (x�; z�; p�; q�) represent a Financial market equilibriumof an economy with Incomplete markets (rank(A) < S). In this economy, theallocation x� is not Constrained Pareto optimal.

Proof. By the decomposition of the budget constraints in the proof ofArrow theorem, this economy is equivalent to one with Complete markets whoseconsumption set at time t = 1 is restricted by

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

12

with the set B(g�t=1(!; z)) de�ned implicitly by266664::

g�s (!s; z)�xis � !is

�::

377775 2< A >; for any i = 1; :::; INote �rst of all that, by construction, p�s 2 g�s (!s; z�): Following the proof ofPareto sub-otimality of Financial market equilibrium allocations, it then follows

that if a Pareto-improving y exists, it must be that

266664::

p�s�yi�s � !is

�::

377775 =2< A >;

for some i = 1; :::; I; while

266664::

g�s (!s; �)�yis � !is

�::

377775 = A�i, for all i = 1; :::; I:

Generic Constrained Pareto sub-optimality of x� follows then directly from thefollowing Lemma, which we leave without proof.9

Lemma 23 Let (x�; z�; p�; q�) represent a Financial Market Equilibrium of aneconomy with rank(A) < S: For a generic set of economies, the constraints266664

::

g�s (!s; z� + dz)

�yis � !is

�::

377775 = A�zi� + dzi

�, for some dz 2 RJIn f0g such

thatXi=1I

dzi = 0; are weakly relaxed for all i = 1; :::; I, strictly for at least

one.10

There is a fundamental di¤erence between incomplete market economies,which have typically not Constrained Optimal equilibrium allocations, and economieswith constraints on the consumption set, which have, on the contrary, Con-strained Optimal equilibrium allocations. It stands out by comparing the re-spective trading constraints

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

9The proof is due to Geanakoplos-Polemarchakis (1986). It also requires di¤erential topol-ogy techniques.10Once again, note that the Lemma implies that a Pareto improving allocation can be found

locally around the equilibrium, as a perturbation of the equilibrium.

13

The trading constraint of the incomplete market economy is determined at equi-librium, while the constraint on the consumption set is exogenous. Another wayto re-phrase the same point is the following. A planner choosing (y; �) will takeinto account that at each (y; �) is typically associated a di¤erent trading con-straint 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:

Remark 24 Consider an economy whose constraints on the consumption setdepend 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 to extendthe analysis of this section to show that this formulation introduces ine¢ cienciesand equilibrium allocations are Constraint Pareto sub-optimal.

Corollary 25 Let (x�; z�; p�; q�) represent a Financial market equilibrium ofa 1-good economy (L = 1) with Incomplete markets (rank(A) < S). In thiseconomy, the allocation x� is Constrained Pareto optimal.Proof. The constraint on the consumption set implied by incomplete mar-

kets, if L = 1, can be written

(xis � !is) = aszi:

It is independent of prices, of the form xit=1 2 B.

Remark 26 Consider an alternative de�nition of Constrained Pareto optimal-ity, due to Grossman (1970), in which constraints 3 are substituted by

30:

266664::

p�s�xi�s � !is

�::

377775 = Azi; for any i = 1; :::; Iwhere p� is the spot market Financial market equilibrium vector of prices. Thatis, the planner takes the equilibrium prices as given. It is immediate to provethat, with this de�nition of Constrained Pareto optimality, any Financial mar-ket equilibrium allocation x�of an economy with Incomplete markets is in factConstrained Pareto optimal, independently of the �nancial markets available(rank(A) � S):

Problem 27 Consider a Complete market economy (rank(A) = S) whose fea-sible set of asset portfolios is restricted by:

zi 2 Z ( RJ ; for any i = 1; :::; I

14

A typical example is borrowing limits:

zi � �b; for any i = 1; :::; I

Are equilibrium allocations of such an economy Constrained Pareto optimal (alsoif L > 1)?

Problem 28 Consider a 1-good (L = 1) Incomplete market economy (rank(A) <S) which lasts 3 periods. De�ne an Financial market equilibrium for this econ-omy as well as Constrained Pareto optimality. Are Financial market equilibriumallocations of such an economy Constrained Pareto optimal?

3 Aggregation

Agent i�s optimization problem in the de�nition of Financial market equilib-rium requires two types of simultaneous decisions. On the one hand, the agenthas to deal with the usual consumption decisions i.e., she has to decide howmany units of each good to consume in each state. But she also has to make�nancial decisions aimed at transferring wealth from one state to the other. Ingeneral, both individual decisions are interrelated: the consumption and port-folio allocations of all agents i and the equilibrium prices for goods and assetsare all determined simultaneously from the system of equations formed by (??)and (??). The �nancial and the real sectors of the economy cannot be isolated.Under some special conditions, however, the consumption and portfolio deci-sions of agents can be separated. This is typically very useful when the analysisis centered on �nancial issue. In order to concentrate on asset pricing issues,most �nance models deal in fact with 1-good economies, implicitly assumingthat the individual �nancial decisions and the market clearing conditions in theassets markets determine the �nancial equilibrium, independently of the indi-vidual consumption decisions and market clearing in the goods markets; thatis independently of the real equilibrium prices and allocations. In this sectionwe shall identify the conditions under which this can be done without loss ofgenerality. This is sometimes 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, then the aggregatespot market demand for the L goods in each state s should must depend onlyon the incomes of the agents in this state (and not in other states) and shouldbe independent of the distribution of income among agents in this state.

Theorem 29 Budget Separation. Suppose that each agent i�s preferences areseparable across states, identical, homothetic within states, and von Neumann-Morgenstern; i.e. suppose that there exists an homothetic u : RL ! R suchthat

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

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

15

Then equilibrium spot prices p� are independent of asset prices q and of the

income distribution; that is, constant inn!i 2 RL(S+1)++

���PIi=1 !

i giveno:

Proof. The consumer�s maximization problem in the de�nition of Finan-cial market equilibrium can be decomposed into a sequence of spot commod-ity allocation problems and an income allocation problem as follows. Thespot commodity allocation problems. Given the current and anticipated spotprices p = (p0; p1; :::; pS) and an exogenously given stream of �nancial incomeyi = (yi0; y

i1; :::; y

iS) 2 RS+1++ in units of numeraire, agent i has to pick a con-

sumption vector xi 2 RL(S+1)+ to

maxui(xi)s:t:p0x

i0 = 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. 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, agent i has to pick a portfolio zi 2 RJ and an income streamyi 2 RS+1++ to

max 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 consump-tion allocation problem into S + 1 �spot market�problems, each of which yieldsthe demands xis(ps; y

is) for each state. By homotheticity, for each s = 0; 1; :::S;

and by identical preferences across all agents,

xis(ps; yis) = y

isxis(ps; 1);

and since preferences are identical across agents,

yisxis(ps; 1) = y

isxs(ps; 1)

Adding over all agents and using the market clearing condition in spot marketss, we obtain, at spot markets equilibrium,

xs(p�s; 1)

IXi=1

yis �IXi=1

!is = 0:

Again by homothetic utility,

xs(p�s;

IXi=1

yis)�IXi=1

!is = 0: (9)

16

Recall from the consumption allocation problem that psxis = yis; for s = 0; 1; :::S:

By adding over all agents, and using market clearing in the spot markets in states,

IXi=1

yis = p�s

IXi=1

xis; for s = 0; 1; :::S (10)

= p�s

IXi=1

!is; for s = 0; 1; :::S:

By combining (9) and (10), we obtain

xs(p�s; p

�s

IXi=1

!is) =IXi=1

!is: (11)

Note how we have passed from the aggregate demand of all agents in the econ-omy to the demand of an agent owning the aggregate endowments. Observe alsohow equation (11) is a system of L equations with L unknowns that determinesspot prices p�s for each state s independently of asset prices q: Note also thatequilibrium spot prices p�s de�ned by (11) only depend !

i throughPI

i=1 !is:

Remark 30 The Budget separation theorem can be interpreted as identifyingconditions under which studying a single good economy is without loss of gen-erality. To this end, consider the income allocation problem of agent i, givenequilibrium 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�) is iden-tical across agents i and, seen as a function of yi, it satis�es the assumptionswe 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 problem becomes:

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�nitionof Financial market equilibrium with l = 1. Note that yis gains the interpretationof agent i�s consumption expenditure in state s, while ws is interpreted as agenti�s income endowment in state s:

17

3.1 The Representative Agent Theorem

A representative agent is the following theoretical construct.

De�nition 31 Consider a Financial market equilibrium (x�; z�; p�; q�) of aneconomy populated by i = 1; :::; I agents with preferences ui : X ! R and endow-ments !i: A Representative agent for this economy is an agent with preferencesUR : X ! R and endowment !R such that the Financial market equilibriumof an associated economy with the Representative agent as the only agent hasprices (p�; q�).

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

Theorem 32 Representative agent. Suppose there exists an homothetic 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, her con-sumption 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 independently ofthe equilibrium of the original economy with I agents, then equilibrium pricescan be read out of the Representative agent�s marginal rates of substitutionevaluated at

PIi=1 !

i. SincePI

i=1 !i is exogenously given, equilibrium prices

are obtained without computing the consumption allocation and portfolio forall agents at equilibrium, (x�; z�):Proof. The proof is constructive. Under the assumptions on preferences in

the statement, we need to show that, for all agents i = 1; :::; I, equilibrium asset

18

prices q� are constant inn!i 2 RL(S+1)++

���PIi=1 !

i giveno.If preferences satisfy

ui(xi) = u(xi0) +PS

s=1 probsu(xis); for all i = 1; ::; I, with an homothetic u; by

the Budget separation theorem, equilibrium spot prices p� are independent of

q� and constant inn!i 2 RL(S+1)++

���PIi=1 !

i giveno: Therefore, p�s!

is 2< A >

is an assumption on fundamentals; in particular on !i: Furthermore, we canrestrict our analysis to the single good economy derived in the previous remark,whose agent i�s optimization problem is:

maxyi2RS+1

++

v(yi)

s:t:

yi0 � w0 = �qzi

yis � ws = aszi; for s = 1; :::S

Write the budget constraints

yi0 � w0 = �qzi

yis � ws = aszi; for s = 1; :::S

as

2664yi0 � w0

:yis � ws

:

3775 2< ��qA � >. Under the homothetic representation of preferencesui(xi), we can show that v(yi) is von Neumann-Morgernstern and

v(yi) = uR(yi0) +SXs=1

probsuR(yis)

for some function uR : R ! R: By Arrow theorem�we can write budget con-straints as

yi0 � wi0 +SXs=1

�s�yis � ws

�= 0

�yis � wis

�Ss=1

2 < A >

But,�wis�Ss=1

2< A > implies that there exist a ziw such that (ws)Ss=1 = Az

iw:

Therefore,�wis�Ss=1

2< A > implies that�yis�Ss=1

= A�zi + ziw

�: We can then

write each agent i�s optimization problem in terms of (yi0�wi0; zi); and the valueof agent i0s endowment is wi0 +

PSs=1 �sw

is = +

PSs=1 �sasz

iw = w

i0 + qz

iw: By

the fact that preferences are identical across agents and by homotheticity of v;then we can write�yi0�q; wi0 + qz

iw

�� wi0

zi�q; wi0 + qz

iw

� �=

�y0�q; wi0 + qz

iw

�� wi0

z�q; wi0 + qz

iw

� �=�wi0 + qz

iw

� �y0 (q; 1)� wi0z (q; 1)

�

19

At equilibrium then�y0 (q

�; 1)� wi0z (q�; 1)

� IXi=1

�wi0 + q

�ziw�=

�y0 �q�;PIi=1 w

i0 + q

�PIi=1 z

iw

��PI

i=1 wi0

z�q�;PI

i=1 wi0 + q

�PIi=1 z

iw

� �=

=

�0

0

�:

and prices q� only depend onPI

i=1 wi0 and

PIi=1 z

iw: But since

�wis�Ss=1

= Aziw;PIi=1 z

iw is a linear translation of

PIi=1 w

i: Finally, let

UR(x) = v(IXi=1

yi) where yis = p�

IXi=1

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

to end the proof.The Representative agent theorem, as noted, allows us to obtain equilib-

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

PIi=1 w

i: Under the assumptions of theRepresentative 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 for pricingassets.

Problem 33 Note that, under the Complete markets assumption, the span re-striction on endowments, p�s!

is 2< A >; is trivially satis�ed. Does the as-

sumption p�s!is 2< A >; for all agents i imply Pareto optimal allocations in

equilibrium.

Problem 34 Assume all agents have identical quadratic preferences. Deriveindividual demands for assets (without assuming p�s!

is 2< A >) and show that

the Representative agent theorem is obtained.

Another interesting but misleading result is the "weak" representative agenttheorem, due to Constantinides (1982).

Theorem 35 Suppose markets are complete (rank(A) = S) and preferencesui(xi) are von Neumann-Morgernstern (but not necessarily identical nor homo-thetic). 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

20

constitutes a Representative agent.

Clearly, then,

q� =SXs=1

MRSRs (!Rs )as;

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

@UR(x)@x0

:

Proof. Consider a Financial market equilibrium (x�; z�; p�; q�). By com-plete markets, the First welfare theorem holds and x� is a Pareto optimal al-location. Therefore, there exist some weights that make x� the solution to theplanner�s problem. It turns out that the required weights are given by

�i =

�@ui(xi�)

@xi�10

��1:

This is left to the reader to check; it�s part of the celebrated Negishi theorem..

This result is certainly very general, as it does not impose identical homo-thetic preferences, however, it is not as useful as the �real�Representative agenttheorem to �nd equilibrium asset prices. The reason is that to de�ne the speci�cweights for the planner�s objective function, (�i)Ii=1; we need to know what theequilibrium allocation, x�; which in turn depends on the whole distribution ofendowments over the agents in the economy.

4 Asset Pricing

Relying on the aggregation theorem in the previous section, in this section wewill abstract from the consumption allocation problems and concentrate on one-good economies. This allows us to simplify the equilibrium de�nition as follows.

4.1 Some classic representation of asset pricing

Often in �nance, especially in empirical �nance, we study asset pricing repre-sentation which express asset returns in terms of risk factors. Factors are tobe interpreted as those component of the risks that agents do require a higherreturn to hold.How do we go from our basic asset pricing equation

q = E(mA)

to factors?

21

4.1.1 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 =Aj

qj. 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), we canrewrite 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)

Emwe have the beta representation of asset prices:

ERj = Rf + �j�m (12)

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 of an assetj is equal to the risk free rate plus the correction for risk, �j�m. Furthermore,we can read (12) as a single factor representation for asset prices, where thefactor is m, that is, if the representative agent theorem holds, her intertemporalmarginal rate of substitution.

4.1.2 Multi-factor beta representations

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

ERj = Rf +FXf=1

�jf�mf(13)

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 .

22

Proposition 36 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 is consis-tent with our basic asset pricing equation when associated to a linear statisticalmodel for the stochastic discount factor m, in the form of m =

PFf=1 bfmf .

Proof. Write 1 = E(mRj) as Rj = Rf � cov(m;Rj)Em and then to substitute

m =PF

f=1 bfmf and the de�nitions of �jf , to have

�mf=var(mf )bfEmf

4.1.3 The CAPM

The CAPM is nothing else than a single factor beta representation of the fol-lowing form:

ERj = Rf + �jf�mf

wheremf = a+ bR

w

the return on the market portfolio, the aggregate portfolio held by the investorsin 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=1 wi0 = w0. Then budget constraints include

xs = Rws (w0 � x0)

Then,

ms = �xs � x#x0 � x#

=�(w0 � x0)(x0 � x#)

Rws ��x#

x0 � x#

23

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 study condi-tional asset market representations, as it implies that the CAPM is intrinsicallya conditional model of asset prices.

4.1.4 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 �(:), the stan-dard 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 sto-

chastic discount factor m which prices asset j. Hansen-Jagannathan are respon-sible for having derived bounds like these and shown that, when the stochasticdiscount factor is assumed to be the intertemporal marginal rate of substitutionof the representative agent (with CES preferences), the data does not displayenough variation in m to satisfy the relationship.

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

m = mp + �

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

Corollary 37 Let (A; q) satisfy No-arbitrage. Then, there exists a unique �� 2<A > such that q = A��:

Proof. By the No-arbitrage theorem, there exists � 2 RS++ such that q =�A: We need to distinguish notationally a matrix M from its transpose, MT :We write then the asset prices equation as qT = AT�T . Consider �p:

�Tp = A(ATA)�1q:

Clearly, qT = AT�Tp ; that is, �Tp satis�es the asset pricing equation. Further-

more, such �Tp belongs to < A >, since �Tp = Azp for zp = (ATA)�1q: Proveuniqueness.We can now exploit this uniqueness result to yield a characterization of the

�multiplicity�of stochastic discount factors when markets are incomplete, and

24

consequently a bound on �(m). In particular, we show that, for a given (q; A)pair a vectorm is a stochastic discount factor if and only if it can be decomposedas a projection on < A > and a vector-speci�c component orthogonal to < A >.Moreover, the previous corollary states that such a projection is unique.

Letm 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 orthogonal projec-tion of m onto < A >, and denote it by mp. We can then write any stochasticdiscount factors m as m = mp+", where " is orthogonal to any vector in < A >;in particular to any Aj . Observe in fact that mp+" is also a stochastic discountfactors since qj = E((mp+ ")aj) = E(mpaj)+E("aj) = E(mpaj), by de�nitionof ". Now, observe that qj = E(mpaj) and that we just proved the uniqueness ofthe stochastic discount factors lying in < A > : In words, even though there is amultiplicity of stochastic discount factors, they all share the same projection on< A >. Moreover, if we make the economic interpretation that the componentsof the stochastic discount factors vector are marginal rates of substitution ofagents in the economy, we can interpret mp to be the economy�s aggregate riskand each agents " to be the individual�s unhedgeable risk.It is clear then that

�(m) � �(mp)

the bound on �(m) we set out to �nd.

25