Alberto Bisin

Lecture Notes on Financial Economics:

Two-Period Exchange Economies

September, 20071

1 Dynamic Exchange Economies

In a two-period pure exchange economy we study financial 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 financial economics,we study

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

sufficient 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 financial marketsare introduced as useful means to characterize financial market equilibria.

1.1 Time and state contingent commodities

Consider an economy extending for 2 periods, t = 0, 1. Let iε{1, ..., I} denoteagents and lε{1, ..., L} physical goods of the economy. In addition, the state ofthe world at time t = 1 is uncertain. Let s ∈ {1, ..., S} denote the state spaceof the economy.

The consumption space is denoted by X ⊆ RL(S+1)+ and define n = L(S+1).

Each agent is endowed with a vector ωi = (ωi0, ωi1, ..., ω

iS), where ωi0εR

L+, and

ωisεRL+, for each s. Let ui : Rn+ −→ R denote agent i’s utility function. We will

assume:

Assumption 1 ωi ∈ Rn++ for all i

Assumption 2 ui is continuous, strongly monotonic, strictly quasiconcave andsmooth, for all i (see Magill-Quinzii, p.50 for definitions and details).2

1Thanks to Francesc Ortega for research assistance.2Note that we are not assuming von Neumann-Morgernstern preferences, that is, an ex-

pected utility representation, for the moment. We will; but it is notationally convenient notto do it right away.

1

Suppose now that at time 0, agents can buy contingent commodities. Thatis, there are contracts that for a price pl promise to deliver 1 unit of good lat time 0, and contracts that for a price pls promise to deliver 1 unit of goodl at time 1 if (and only if) state s occurs. Denote by xi = (xi0, x

i1, ..., x

iS) the

vector of all such contingent commodities purchased by agent i at time 0, wherexi0 ∈ RL+, and xis ∈ RL+,

Denote by p0 the vector (p10, . . . , pL0) and by ps the vector (p1s, . . . , pLs).Under the assumption that the markets for all the goods at all time periods

are open at time 0, agent i’s budget constraint can be written as3

p0(xi0 − ωi0) +S∑s=0

ps(xis − ωis) = 0 (1)

The interpretation of these markets is that at time 0, agents can buy unitsof a contract that promises (with no shade of doubt) to deliver one unit of goodl in period 1 in state s for a price of plts.

Definition 1 (x, p) is an Arrow-Debreu Equilibrium ifi. For all iε{1, ..., I},

xiε arg maxui(xi) s.t.p0(xi0 − ωi0) +

∑Ss=0 ps(x

is − ωis) = 0

.

ii. For all lε{1, ..., L} and all sε{1, ..., S},I∑

i=1

(xil0 − ωil0) = 0.

I∑

i=1

(xils − ωils) = 0.

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

or at the same time t but different state s′. 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.4

3We write the budget constraint with equality. This is without loss of generality undermonotonicity of preferences, an assumption we shall maintain.

4Having set definitions for 2-periods Arrow-Debreu economies, it should be apparent how ageneralization to any finite T -periods economies is in fact effectively straightforward. Infinitehorizon will be dealt with in successive notes.

2

Definition 2 Let (x, p) be an Arrow-Debreu Equilibrium. We say that x =(x1, . . . , xI) ∈ RnI is Pareto Optimal if there does not exist an allocation y =(y1, . . . , yI) ∈ RnI such that

i yi is preferred to xi for all i and strictly preferred for some i,

ii∑Ii=1(yis − ωis) = 0, for s=0,1,...,S,

Theorem 1 First Welfare Theorem. Any Arrow-Debreu Equilibrium allocationis Pareto Optimal.

Proof. By contradiction. Suppose an y = (y1, . . . , yI) ∈ RnI such that

i yi is preferred to xi for all i and strictly preferred for some i,

ii∑Ii=1(yis − ωis) = 0, for s=0,1,...,S,

exists. Then, by (i) of the definition of Arrow-Debreu equilibrium, it mustbe that

p0(yi0 − ωi0) +S∑s=0

ps(yis − ωis) ≥ 0,

for all i; and

p0(yi0 − ωi0) +S∑s=0

ps(yis − ωis) > 0,

for at least one i.Summing over i, then

p0

I∑

i=1

(yi0 − ωi0) +S∑s=0

ps

I∑

i=1

(yis − ωis) > 0

which contradicts requirement (ii) above when prices are strictly positive (whichis in turn implied by strong monotonicity).

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 in assetsjε{1, ..., J}.

Let the payoff vector of asset j be denoted by aj = (aj1, ..., ajS), where ajs

are the units of good l = 1 (numeraire) that asset j pays in state s, i.e., we areassuming that p0s = 1, for all s. To summarize the payoffs of all the availableassets, define the S × J asset payoff matrix

A =

a11 ... aJ1... ...a1S ... aJS

.

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It will be convenient to define As to be the s-th row of the matrix. Note thatit contains the payoff of each of the assets in state s.

Let xisl denote the amount of physical good l that agent i consumes in goods. Let p = (p0, p1, ..., pS), where ps ∈ RL+ for each s, denote the spot pricevector for physical goods. Let q = (q1, ..., qJ) ∈ RJ , denote the prices for theassets. Quantities will be row vectors and prices will be column vectors. Notethat the prices of assets can be negative, as we did not impose non-negativityon the assets’ payoffs.

Given prices (p, q) and the asset structure A, agent i picks a consumptionvector xi ∈ RL(S+1) and a portfolio z ∈ RJ to

maxui(xi)s.t.p0(xi0 − ωi0) + qzi = 0ps(xis − ωis) = Asz

i, for s = 1, ...S.

The equilibrium concept for this economy is the following.

Definition 3 (x, z; p, q) is a Financial Markets Equilibrium ifi. for all i ∈ I,

xi ∈ arg maxui(xi)s.t.p0(xi0 − ωi0) + qzi = 0ps(xis − ωis) = Asz

i, for s = 1, ...S,

ii. and markets clear

I∑

i=1

(xis − ωis) = 0, for s=0,1,...,S (2)

I∑

i=1

zi = 0. (3)

Observation 1. The economy just introduced is characterized by assetmarkets in zero net supply, that is, no endowments of assets are allowed for. Infact it is straightforward to extend the analysis to assets in positive net supply.Part of each agent i’s endowment (to be specific: the projection of his/herendowment on the asset span, < A >= {τ ∈ RS : τ = Az, z ∈ RJ}) can berepresented as an asset endowment, ziw. As a consequence, each agent’s budgetset can be written as

p0(xi0 − ωi0) + q(zi − ziw

)= 0

ps(xis − ωis) = Aszi, for s = 1, ...S,

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and market clearing becomes:

I∑

i=1

(xi0 − ωi0) = 0, for s=1,...,S

I∑

i=1

(xis − ωis −Asziw) = 0, for s=1,...,S

I∑

i=1

zi − ziw = 0.

The definition of financial market equilibrium as well as the whole analysis thenproceeds unchanged.

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 financial 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.

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 financial markets equilbria.

For convenience, define the (S + 1)× J matrix

W =[ −qA

].

Definition 4 (No-Arbitrage Condition) We will say that W satisfies the No-Arbitrage Condition if there does not exist z ∈ RJ such that Wz > 0, i.e.all components of the (S + 1) vector Wz are non-negative and at least one ispositive.

The No-Arbitrage condition can be equivalently formulated in the followingway. Define the span of W to be

< W >= {τ ∈ RS+1 : τ = Wz, z ∈ RJ}.Note that this set contains all the feasible wealth transfers with the given assetstructure A. Now, we can say that W satisfies the No-Arbitrage condition iff

< W >⋂RS+1

+ = {0}.

Clearly, requiring that W = (−q,A) satisfies the No-Arbitrage condition is amilder requirement than equilibrium. Given that we assumed strong monotonic-ity in preferences, No-Arbitrage is equivalent to requiring the agent’s problem

5

to be well defined. The next result is remarkable since it provides a founda-tion for asset pricing with milder requirements than imposing equilibrium inthe economy.

Theorem 1 (No-Arbitrage theorem)

< W >⋂RS+1

+ = {0} ⇐⇒ ∃π̂ ∈ RS+1++ such that π̂W = 0.

First, observe that there is no uniqueness claim on the π̂, just existence isclaimed. Next, notice how π̂W = 0 provides a pricing formula for assets:

π̂W =

...

−π̂0qj + π̂1a

j1 + ...+ π̂Sa

jS

...

=

...0...

Jx1

and, rearranging, we obtain for each asset j,

qj =π̂1

π̂0aj1 + ...+

π̂Sπ̂0ajS , (4)

or defining πs = π̂1π̂0, for s=1,...S, we obtain

qj = π1aj1 + ...+ πSa

jS . (5)

Note how the positivity of all components of π̂ was necessary to obtain (4).Proof.=⇒Define the simplex in RS+1

+ as ∆ = {τ ∈ RS+1+ :

∑Ss=0 τs = 1}. Note that

by No-Arbitrage condition, < W > intersection ∆ is empty. The proof hingescrucially on the following separating result, which we shall take without proof..

Lemma. Let X be a finite 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, suppose K and M are disjoint. Then, there existsπ̂ ∈ X\{0} such that

supτ∈M

π̂τ < infτ∈K

π̂τ.

Let X = RS+1+ , K = ∆ and M =< W >. Observe that all the required

properties hold and so the lemma applies. As a result, there exists π̂ 6= 0 suchthat

supτ∈<W>

π̂τ < infτ∈∆

π̂τ. (6)

Let us now show π̂ ∈ RS+1++ . Suppose, on the contrary, that there is some s

for which π̂s ≤ 0. Then note that in (6 ), the RHS≤ 0. By (6), LHS < 0 but thiscontradicts the fact that 0 ∈< W > which implies that RHS is non-negative.

We still have to show that π̂W = 0, or in other words, that π̂τ = 0 for allτ ∈< W >. Suppose, on the contrary that there exists τ ∈< W > such thatπ̂τ 6= 0. Since < W > is a subspace, there exists α ∈ R such that ατ ∈< W >

6

and π̂ατ is as large as we want. However, RHS is bounded above, which impliesa contradiction.⇐=Suppose that there exists π̂ ∈ RS+1

++ such that π̂W = 0, or q = πA. Fromthe agent’s problem, recall the budget constraint

p0

(xi0 − ωi0

)+ qzi = 0

ps(xis − ωis

)= Asz

i, for s = 1, ...S.

Plugging in q = πA and expanding the first equation yields

p0

(xi0 − ωi0

)+∑Ss=1 πsps(x

is − ωis) = 0

ps(xis − ωis) = Aszi, for s = 1, ...S.

(7)

Observe now that the set defined in the first equation in (7) is a compact set(since it is a standard Arrow-Debreu budget set). Note further that the secondequation defines a closed set. Both taken together are a closed subset of acompact set and therefore it itself is compact. The compactness of the budgetset defined by (7) implies that the agent’s problem achieves a maximum levelof utility. This, in turn, implies that no arbitrage opportunities were availableto her. QED.

Three final remarks to this section.Observation 2. An asset which pays one unit of numeraire in state s and

nothing in all other states (Arrow security), has price πs according to (4). Suchasset is called Arrow security.

Observation 3. Should we expect to find a unique π vector ? Or, if thereare many, how many? To address this question, notice how we can express thesystem of equation given by (5) in a simplified manner. Note that it is a systemof J equations with S unknowns. Define the set of solutions to that system as

R(q) = {π ∈ RS++ : q = πA}.

Suppose the matrix A has maximum rank, that is it has rank J (yet another wayto say this is that it has linearly independent column vectors). In general, thenR(q) will have dimension S−J . In particular, if we had S linearly independentassets, the solution set has dimension zero, and there is a unique π vector thatsolves (5).

Observation 4. Let ms = πsprobs

. Then

qj = E (maj)

We conclude with the following important definition:

Definition 5 An economy has Complete Financial Markets if rank(A) = S.

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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-fledged economy. Among other things, this will allow us toprovide some economic content to the vector π

Recall the definition of equilibrium for a one-good economy with financialmarkets. By taking the First Order Conditions (necessary and sufficient for amaximum under our assumptions) with respect to zij of the individual problemfor an arbitrary price vector q, we obtain that

qj =S∑s=1

probsMRSis(xi)ajs = E

(MRSAj

), for all j = 1, ..., J. (8)

where MRSis(xi) is 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) =

∂u(xis)

∂xis1∂u(xi0)

∂xi01

That is, at any solution of the agent’s problem, the marginal cost of one moreunit of asset j (qj) is equalized to the marginal valuation of that agent for theasset’s payoff.

Let(. . .MRSis . . .

)denote the vector of marginal rates of substitution for

agent i, an S dimentional (strictly positive, under Assumption A.2) vector.Compare equation (8) to the previous (4). Clearly, at any equilibrium, con-

dition (8) has to hold for each agent i. Therefore, in equilibrium, the vectorof marginal rates of substitution of any arbitrary agent i can be used to priceassets; that is any of the agents’ vector of marginal rates of substitution (nor-malized by probabilities) solves equation (4) when substituted to π.

Another way to state the same observation is that each vector MRSi (nor-malized by probabilities) belongs to R(q). But we noticed before that whenmarkets are complete, that is when the rank of A is S, the set R(q) is in fact asingleton. In this case the it must be that MRSi is equalized across agents i atequilibrium.

Problem 1 Write the Pareto problem for the economy and show that at anyPareto optimal allocation, the MRSi = MRS for all agent i. Also show that anallocation which satisfies the feasibility conditions (market clearing) for goodsand such that MRSi = MRS for all agent i is Pareto optimal.

We conclude that when markets are complete equilibrium allocations arePareto optimal.

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Problem 2 (Economies with bid-ask spreads) Extend our economy by assumingthat whenever an agent is buying any asset j, the price she pays is qj + γj , withγj > 0 and exogenous. Whereas, when the agent is shortselling asset j, the priceshe charges is qj. First, write the budget constraint. Second, write the FOCfor the agent’s problem, in particular the conditions for assets demand/supply.Third, derive a generalized asset pricing relation (not an equation, is it?) thatrelates MRS to asset prices.

1.5 Arrow theorem

There is an apparently different equilibrium concept that can be used withour economy with financial markets, {u, ω,A}. It was essentially introduced byArrow (1952) and focuses on feasible wealth transfers across states directly, withno explicit mention to the specific financial markets. Let the span of A, < A >be defined as:

< A >={τ ∈ RS : τ = Az, z ∈ RJ}

Definition 6 (x, p, π) is said to be a No-Arbitrage Equilibrium for the economycharacterized by {u, ω,A} if

i. for all i ∈ I,

xi ∈ arg maxui(xi)s.t.

(xi0 − ωi0) +∑Ss=1 πsps(x

is − ωis) = 0

ps(xis − ωis) ∈< A >, for s = 1, ...S,

ii. and markets clear

I∑

i=1

(xis − ωis) = 0, for s=0,1,...,S and for all l =1, . . . , L (9)

Note that when matrix A has rank S, < A >= RS and the No-Arbitrageequilibrium reduces to the standard Arrow-Debreu equilibrium. We thereforecan apply the First Welfare Theorem in its basic form to conclude that completemarkets imply Pareto efficiency (note that this is a different proof of the state-ment above, reached via equalization of marginal rates of substitution acrossagents).

Proposition 2 For any economy with financial markets {u, ω,A},the set ofFinancial markets equilibria is equivalent to the set of No-Arbitrage equilibria.

The proof is straightforward, given the construction of No-arbitrage equi-libria and is left to the reader.

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2 Constrained Pareto Optimality

Under complete markets, the First Welfare Theorem holds. However, underIncomplete Markets Financial Markets Equilibria are generically inefficient in aPareto sense. That is, a planner could find an allocation that improves someagents without making any other agent worse off. To be more precise, considerthe following:

Definition 7 Let (x, z; q) be a Financial Market equilibrium. We say that x isa Pareto Optimal allocation if there does not exist {(yi, θi) : i = 1, ..., I} suchthat

i yi is preferred to xi for all i and strictly preferred for some i,

ii∑Ii=1(yis − ωis) = 0, for s=0,1,...,S,

The following result is relatively straightforward:

Theorem 2 A Financial Market Equilibrium of an incomplete market econ-omy, that is, of an economy with rank(A) < J , generates allocations that aregenerically5 not Pareto Optimal.

While the proof is simple, it requires techniques of differential topology (asany time statements hold generically). An intuition is however straighforward:think of reproducing the proof of the First Welfare Theorem, but requiring thatPareto Optimal allocations need only satisfy

I∑

i=1

(yis − ωis) = 0, for s = 0, 1, ..., S

while equilibrium allocation need also satisy

ps(yis − ωis) = Asθi, for all i and all s.

It is obvious that, unless the constraint ps(yis−ωis) = Asθi, for all i and all s, is

not binding at an equilibrium (here’s why the statement only holds generically),the planner can improve on equilibrium allocations.

The problem is that markets are assumed incomplete exogenously. There isno reason in the fundamentals of the model why they should be. But supposethat for whatever reason markets are incomplete, the planner is subject to thesame set of constraints on wealth transfers that incomplete markets imposeon agents in a competitive equilibrium. Recall that the feasible wealth vectorsacross states are given by the span of the payoff matrix. That can be interpretedas the economy’s “financial technology” and it seems reasonable to impose thesame technological restrictions on the planner’s reallocations. The formalizationof an efficiency notion capturing this idea follows.

5We say that a statement holds generically when it holds for a full Lebesgue-measuresubset of the parameter set which characterizes the economy, that is, in our case the productof the compact space of endowments and the compact space of two-parameter parametrizationof utility functions for each agent; see Magill-Shafer, ch. 30 in W. Hildenbrand and H.Sonnenschein (eds.), Handbook of Mathematical Economics, Vol. IV, Elsevier 1991.

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Definition 8 (Diamond, 1968; Geanakoplos-Polemarchakis, 1986) Let (x, z; p, q)be a Financial Market equilibrium. We say that x is a Constrained Pareto Op-timal allocation if there does not exist {(yi, θi) : i = 1, ..., I} such that

i. yi is preferred to xi for all i and strictly preferred for some i,ii.∑Ii=1(yis − ωis) = 0, for s = 0, 1, ..., S, and l = 1, 2, , ..., L

iii. ps(yis−ωis) = Asθi, for all i and s, where ps is a spot market equilibrium

price vector when agent i is endowed with income Asθi.

Once again, we shall be content with an intuition (but a powerful one) for theresult. Consider an economy with complete markets, that is, rank(A) = J,butsubject to the following constraints on the consumption set:

xi ∈ B ( X, for all i

(it is not easy to find a meaningful economic example of constraints directly on

consumption; but it is still useful to consider this economy to better understandthe issue of constrained optimality) It is straightforward to see that the budgetconstraints facing the agents in this economy, using the No-arbitrage prices as inthe proof of the Arrow Theorem, can be reduced into a single present-discountedbudget constrained of the form:

p0(xi0 − ωi0) +S∑s=1

πsps(xis − ωis) = 0

For this economy the natural notion of Constrained Optimality is the onein the definition above with yi ∈ B ( X, for all i instead of condition iii).Once again, try and reproduce the proof of the First Welfare Theorem requiringthat Pareto Optimal allocations need also satisfy xi ∈ B ( X, for all i. It isstraighforward to see that the proof goes trough with no changes. The economywe just introduced therefore constrained optimal equilibrium allocations.

But consider now the economy with incomplete markets. We have the fol-lowing:

Theorem 3 ( Geanakoplos-Polemarchakis, 1986) For the L-good economy, Fi-nancial Market Equilibria generate allocations that are generically not Con-strained Pareto Optimal.

What is the fundamental difference between incomplete market economies(which, according to the Theorem, have typically not Constrained Optimal equi-librium allocations) and economies with constraints on the consumption set(whic, we have shown, have on the contrary Constrained Optimal equilibriumallocations)? To answer the question let’s compare the respective trading con-straint which both the planner and the agents face:

ps(xis−ωis) = Asθi, for all i and s , where ps ......... vs. xi ∈ B ( X, for all i

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One fundamental difference stands out: the trading constraint of the incompletemarket economy involves prices, that is, it’s endogenous, while the constrainton the consumption set does not. We claim this is the driving force behindthe constrained inefficiency result. To convince yourself of this, try try andreproduce the proof of the First Welfare Theorem for the incomplete marketeconomy. Requiring that Pareto Optimal allocations also satisfy ps(yis − ωis) =

Asθi, for all i and s, is meaningless without taking a stand on the prices ps at

which the constraint is evaluated. If the prices are the same at which the agentsevaluate the constraint, that is, the equilibirum prices ps, then in fact equilibriaare constrained efficient. But in general prices ps are different from ps as theyare required to clear spot markets in which agents have different wealth levels,that is Asθi as opposed to Asz

i.But then the constraint on Pareto Optimalallocations is different than that on financial market trading faced by agents atequilibrium.

Another way to re-phrase the same point is the following: a planner choosing{(yi, θi) : i = 1, ..., I} will take into account that at each {(yi, θi) : i = 1, ..., I}is typically associated a different trading constraint ps(xis−ωis) = Asθ

i, for all iand s; while any agent i will choose (xi, zi) to satisfy ps(xis−ωis) = Asz

i, for all staking as given the constraint (by perfect competition the agent is strategicallysmall).

Consider however a 1-good economy. In this case the trading constraint onfinancial market takes the form:

(xis − ωis) = Asθi

and no prices are involved. As a consequence, in this case it follows from theargument above that equilibrium allocations are Constrained Optimal.

Theorem 4 (Diamond, 1968) For the 1-good economy, Financial Market Equi-libria generate allocations that are Constrained Pareto Optimal.

Diamond provided a different proof of this result.Proof. Let us start by providing yet a third equivalent equilibrium definition

for the one-good economy with financial markets. It consists of substituting inthe constraints in the agents’ problems and take derivatives with respect to theportfolio assets. For simplicity, assume vNM utility. Define zi0 = xi0 − ωi0. Notethat (x, z; q) is a Financial Markets equilibrium if and only if,

i. for all i,

zi = (zi0, zi) ∈ arg maxzi∈RJ+1{u(ωi0 + zi0) +

∑Ss=1 probsu(ωis +Asz

i)}s.t.zi0 + qzi = 0,

ii. and∑Ii=1 z

ij = 0, for j = 0, 1, ..., J.

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Note that this looks like an Arrow-Debreu equilibrium but with an unre-stricted domain for the choice variables (which poses some difficulties for equi-librium existence that we shall ignore here). Suppose that (x, z; q) is a Fi-nancial Markets Equilibrium. In addition, assume that q ≥ 0, which is with-out loss of generality if we redefine the payoff matrix. Suppose that this isnot a Constrained Pareto Optimal allocation. That is, suppose there exists{(θi0, θi) : i = 1, ..., I} such that all agents are at least as well off as in theequilibrium but there exists an agent i that is strictly better off. By utilitymaximization, it must be true that

θi0 + qθi > 0.

Adding now budget constraints over all agents, we obtain

I∑

i=1

θi0 + q

I∑

i=1

θi > 0,

which contradicts market clearing in the asset markets. QED.

Problem 3 Consider an otherwise complete markets economy which is thoughsubject to another set of trading constraints in financial markets:

θi ∈ Θ ( RJ

(for example, a set of limited short sales constraints: θi ≤ θ). Are equilibrium

allocations of such an economy Constrained Optimal? [Distinguish the case inwhich the economy is 1-good from the case in which it is a L-goods economy.]

Problem 4 Consider a 1-good incomplete markets which lasts 3 periods. De-fine an Financial Market equilibrium for this economy as well as ConstrainedOptimality. Are equilibrium allocations of such an economy Constrained Opti-mal?

3 Aggregation

As can be seen in the agents’ problem above, utility maximization requiresmaking two types of decisions simultaneously. On the one hand, each 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 now, on top of that, shehas to make financial decisions aimed at transferring wealth from one state tothe other. By inspection of the equilibrium definition, we see that, in general,both individual decisions are interrelated. And, moreover, the consumptionallocation, portfolio allocation and the equilibrium prices for goods and assetsare all determined simultaneously from the system of equations formed by (4)and (4). In words, the financial and the real sectors of the economy cannot beisolated.

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In order to concentrate on asset pricing issues, most finance models deal with1-good economies, implicitly assuming that the individual financial decisionsand the market clearing conditions in the assets markets determine the financialequilibrium, independently of the individual consumption decisions and marketclearing in the goods markets, that is independently of the real equilibrium pricesand allocation. Under which conditions can the financial and real equilibriumproblem be separated? This is sometimes called ”the problem of aggregation.”

The object of this section is to derive conditions under which a ‘financeeconomy’ is generated from an equilibrium of the overall economy when thespot prices of the goods are fixed at their equilibrium values. The idea is thefollowing. If we want equilibrium prices on the spot markets to be independentof equilibrium on the financial markets, then the aggregate spot market demandfor the L goods in each state s should depend only on the incomes of the agents inthis state (and not in other states) and should be independent of the distributionof income among agents in this state.

The consumer’s maximization problem can be decomposed into a sequenceof spot commodity allocation problems and an income allocation problem asfollows.

The spot commodity allocation problems. Given the current and an-ticipated spot prices p = (p0, p1, ..., pS) and an exogenously given stream offinancial income yi = (yi0, y

i1, ..., y

iS) in units of numeraire, agent i has to pick a

consumption vector xi ∈ RL(S+1)+ to

maxui(xi)s.t.p0x

i0 = yi0

psxis = yis, for s = 1, ...S.

Let the S + 1 demand functions be given by xis(p, yi), for s = 0, 1, ...S. Definenow the indirect utility function for income by

vi(yi; p) = ui(xi(p, yi)).

The Income allocation problem. Given the prices (p, q), ωi, and theasset structure A, agent i has to pick a portfolio zi ∈ RJ and an income streamyi ∈ RS+1 to

max vi(yi; p)s.t.p0ω

i0 − qzi = yi0

psωis +Asz

i = yis, for s = 1, ...S.

Note how the income streams that are available to agent i are constrained bythe existing asset structure.

Proposition 1 Budget Separation/Aggregation. Suppose that agents’ pref-erences are separable across states, identical and homothetic within states, and

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von Neumann-Morgenstern, i.e. suppose that there exists an homothetic u :RL −→ R such that

ui(xi) = u(x0) +S∑s=1

probsu(xs), for all i,

where probs is the objective probability of state s. Then,

xis(p, yi) = xis(ps, y

is) for s=0,1,...,S.

Moreover, p can be solved for independently from q and the economy’s incomedistribution {yi, }i∈I .

Proof. By additive separability across states of the utility, we can breakthe consumption allocation problem into S + 1 ‘spot market’ problems, each ofwhich yields the demands xis(ps, yis) for each state. By homotheticity, for eachs = 0, 1, ...S, and by identical preferences across all agents,

xis(ps, yis) = yisx

is(ps, 1) = yisxs(ps, 1)

Now, adding over all agents and using the market clearing condition in spotmarkets s, we obtain

xs(ps, 1)I∑

i=1

yis −I∑

i=1

ωis = 0.

Again by homothetic utility,

xs(ps,I∑

i=1

yis)−I∑

i=1

ωis = 0. (10)

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 mar-kets in state s,

I∑

i=1

yis = ps

I∑

i=1

xis, for s = 0, 1, ...S (11)

= ps

I∑

i=1

ωis, for s = 0, 1, ...S.

By combining (10) and (11), we obtain

xs(ps, psI∑

i=1

ωis) =I∑

i=1

ωis. (12)

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 also

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how equation (12) is a system of L equations with L unknowns that determinesspot prices for state s regardless of the asset structure or the asset prices.

Furthermore, suppose that we have obtained the spot equilibrium prices fromequation (12), denote them by p. At that point, we can solve for the equilibriumasset prices by solving the individual problems

max vi(yi; p)s.t.p0ω

i0 − qzi = yi0

psωis +Asz

i = yis, for s = 1, ...S

for each agent i, and combining them with the market clearing equations forassets.

Finally, note that the indirect utility vi is identical across agents i and, seenas a function of yi, it satisfies the assumptions we have imposed on u as afunction of xi, in Assumption A.2.

3.1 The Representative Agent Theorem

We want to know when the equilibrium asset prices of a 1-good economy withheterogeneous agents coincide with the equilibrium prices of a similar econ-omy with only when agent, whose endowments are the aggregate endowmentsof the original economy. This assumption is behind much of the empiricalmacro/finance literature.

Theorem 3 Define < A >= {τ ∈ RS : τ = Az, z ∈ RJ}. Suppose thatωi ∈< A >, for all agents i. Suppose also that all agents share the sameidentical and homothetic preferences. Then the equilibrium asset prices dependonly on the aggregate endowments at each state, but not on the distribution ofthese endowments across individuals.

Note that under the Complete Markets assumption, the span restriction onendowments is trivially satisfied. More interestingly, observe that the “aggregateagent” does not trade in the asset markets in equilibrium (since he is alone!).This is the Lucas’ trick for pricing assets.

Proof. Since ωi ∈< A >, there exists ziw for each i such that ωi = Aziw.It follows that ωs + Asz

i = As(ziw + zi) across all states and agents. Thisequation together with the reformulation of the financial markets equilibrium inthe previous proof of CPO leave us with an Arrow-Debreu economy where agentsderive utility from portfolios. Keeping that in mind and using the identicalhomotheticity assumption, we get that individual asset demands can be writtenas

zi = z(q, 1)(ziw0+ qziw),

where the second component is individual i’s wealth. Equilibrium asset pricesare determined by

I∑

i=1

zi = z(q, 1)I∑

i=1

(ziw0+ qziw) = 0.

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Exercise 3. Does the assumption ωi ∈< A >, for all agents i imply Paretooptimal allocation in equilibrium.

Exercise 4. Assume all agents have identical quadratic preferences, don’tmake the span restriction on individual endowments and derive individual de-mands for assets. Note how they are linear in wealth. Observe also that theRepresentative Agent result is obtained for these preferences too.

Another interesting but misleading result is the ”weak” representative agenttheorem, due to Constantinides (1988).

Theorem 4 Suppose markets are complete and preferences are vNM, denotedby U i(xi) and not necessarily identical nor homothetic. Let (x, q) be a Financial

Markets Equilibrium. Define now θi =(∂ui(xi)∂x0

)−1

, for all i. Then, q can beread off the MRS of a representative agent with preferences

W (x1, ..., xI) =I∑

i=1

θiU i(xi)

evaluated at the solution of the problem

maxx1,...,xI W (x1, ..., xI)s.t.∑Ii=1(xis − ωis) = 0, for s = 0, 1, ...S.

Proof. Consider equilibrium (x, q). By complete markets, the First Funda-mental Theorem holds and x is a Pareto optimal allocation. Therefore, thereexist some weights that make x the solution to the planner’s problem. It turnsout that the required weights are given by

θi :=(∂ui(xi)∂x0

)−1

.

(This is left to the reader to check.)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 find equilibrium asset prices. The reason is that to definethe specific weights for the planner’s objective function, we need to know whatis the equilibrium allocation, which in turn depend 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 definition as follows.

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4.1 Some classic representation of asset pricing

Often in finance, especially in empirical finance, 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

qj = E(mA)

to factors?

4.1.1 Single factor beta representation

Consider the basic asset pricing equation

qj = E(mAj)

Let the return on asset j, Rj , be defined as Rj = Ajqj . Then the asset pricing

equation becomes1 = E(mRj)

(note also that, this equation applied to the risk free rate, denoted Rf , becomesRf = 1

Em . Using the fact that for two random variables x and y, E(xy) =ExEy + cov(x, y), we can rewrite it as:

ERj =1Em

− cov(m,Rj)Eπ

= Rf − cov(m,Rj)Em

Finally, letting

βj =cov(m,Rj)var(m)

and

λπ = −var(m,Rj)

Em

we have the beta representation of asset prices:

ERj = Rf + βjλm (13)

We interpret βj as the ”quantity” of risk in asset j and λm (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, βjλm. Furthermore,we can read (13) as a single factor representation for asset prices, where thefactor is m, that is, if the representative agent theorem holds, his intertemporalmarginal rate of substitution.

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4.1.2 Multi-factor beta representations

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

ERj = Rf +F∑

f=1

βjfλmf (14)

where mf is some random variable, normalized by E(mf ) = 0, which takes theinterpretation of risk factor and

βjf =cov(mf , R

j)var(mf )

is its beta, the loading of the return on the factor.It is not hard to see that such a representation is equivalent to (implies and

is implied by):

m = Rf +F∑

f=1

bfmf and 1 = E(mRj)

In other words, a multi-factor beta representation for asset returns is consistentwith our basic asset pricing equation when associated to a linear statisticalmodel for the stochastic discount factor m, in the form of m = Rf+

∑Ff=1 bfmf .

The proof just requires to write 1 = E(mRj) as Rj = Rf − cov(m,Rj)Em and

then to substitute m = Rf +∑Ff=1 bfmf and the definitions of βjf , to have

λmf = −var(mf )bEmf

4.1.3 The CAPM

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

ERj = Rf + βjfλmf

where mf is Rw the return on the market portfolio, the aggregate portfolio heldby the investors in the economy.

It can be easily derived from an equilibrium model under special assump-tions.

For example, assume preferences are quadratic:

u(xo, x1) = −12

(x− x∗)2 − 12

∑s

probs(xs − x∗)2

Moreover, assume agents have no endowments at time t = 1. Then budgetconstraints include

xs = Rws (w0 − x0)

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(e.g., think of Rw =∑j θjR

j , where θj is the fraction of aggregate investmentson asset j).

Then,

ms = β(xs − x∗)(x0 − x∗) =

β(w0 − x0)x0 − x∗) Rws −

βx∗

x0 − x∗which is the CAPM.

Note however that β(w0−x0)x0−x∗) and βx∗

x0−x∗ are not constant as they do dependon equilibrium allocations. This will be important when we study conditionalasset market representations, as it implies that the CAPM is intrinsically aconditional model of asset prices.

4.1.4 Bounds on stochastic discount factors

Write beta representation of asset returns as:

ERj −Rf = −cov(m,Rj)Em

= −ρ(m,Rj)σ(m)σ(Rj)Em

where ρ(., .) denote the correlation coefficient and σ(.), the standard deviation.Noticing that 0 ≤ ρ(m,Rj) ≤ 1, and ERj −Rf ≥ 0 we have

| ERj −Rfσ(Rj)

|≤ σ(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 ofsubstitution of the representative agent (with CES preferences), the data doesnot display enough 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 payoffs, denoted mp,with the property that any other stochastic discount factor m satisfies:

m = mp + ε

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

Define first< A >= {τ ∈ RS : τ = Az, z ∈ RJ}.

Corollary 5 Let (A, q) satisfy the No-Arbitrage condition, where A is the SxJpayoff matrix and q is the J-dimensional vector of asset prices. Then, thereexists a unique τ∗ ∈< A > such that q = A′τ∗.

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Proof. By the NA theorem, (A, q) satisfying the NA condition implies thatthere exists π ∈ RS++ such that q′ = A′π′. Furthermore, there exists a uniqueπ′ in the span of A:

π′ = A(A′A)−1q.

Note that such π′ belongs to < A >, since π′ = Az∗ for z∗ = (A′A)−1q.We can now exploit this uniqueness result to yield a characterization of the

“multiplicity” of stochastic discount factors (SDF) when markets are incom-plete, and consequently a bound on σ(m). In particular, we show that, for agiven (q ,A) pair, an Sx1 vector is an SDF if and only if it can be decomposedas a projection on < A > and a vector-specific component orthogonal to < A >.Moreover, the previous corollary states that such a projection is unique.

Let m ∈ RS++ be any SDF (that is a π rescaled by the state probabilities:for any s = 1, . . . , S, ms = πs

probsfor some π) for (q, A), i.e. qj = E(mAj), for

j = 1, ..., J. Consider the orthogonal projection of m onto < A >, and denote itby mp. We can then write any SDF m as m = mp + ε, where ε is orthogonalto any vector in < A >, in particular to any Aj . Observe in fact that (mp + ε)is also a SDF since qj = E((mp + ε)aj) = E(mpaj) + E(εaj) = E(mpaj),by definition of ε. Now, observe that qj = E(mpAj) and that we just provedthe uniqueness of the SDF lying in < A > . In words, even though there is amultiplicity of SDFs, they all share the same projection on < A >. Moreover, ifwe make the economic interpretation that the components of the SDFs vectorare MRSs of agents in the economy, we can interpret mp to be the economy’saggregate risk and each agents ε to be the individual’s unhedgeable risk.

It is clear then thatσ(m) ≥ σ(mp)

the bound on σ(m) we set out to find.

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