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Econometric Society

The Consumption-Based Capital Asset Pricing Model

Author(s): Darrell Duffie and William Zame
Source: Econometrica, Vol. 57, No. 6 (Nov., 1989), pp. 1279-1297
Published by: Econometric Society
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Econometrica,Vol. 57, No. 6 (November, 1989), 1279-1297



The paper provides conditions on the primitives of a continuous-time economy under

which there exist equilibria obeying the Consumption-BasedCapital Asset Pricing Model
(CCAPM). The paper also extends the equilibrium characterization of interest rates of Cox,
Ingersoll, and Ross (1985) to multi-agent economies. We do not use a Markovian state
KEYwoRDs: Finance, general equilibrium theory, asset pricing theory, continuous-time
stochastic processes, CAPM.


THIS WORK PROVIDESsufficient conditions on agents' primitives for the validity

of the Consumption-Based Capital Asset Pricing Model (CCAPM) of Breeden
(1979). As a necessary condition, Breeden showed that in a continuous-time
equilibrium satisfying certain regularity conditions, one can characterize returns
on securities as follows. The expected "instantaneous" rate of return on any
security in excess of the riskless interest rate (the security's expected excess rate of
return) is a multiple common to all securities of the "instantaneous covariance"
of this excess return with aggregate consumption increments. This common
multiple is the Arrow-Pratt measure of risk aversion of a representative agent.
(Rubinstein (1976) published a discrete-time precursor of this result.) The exis-
tence of equilibria satisfying Breeden's regularity conditions had been an open
issue. We also show that the validity of the CCAPM does not depend on
Breeden's assumption of Markov state information, and present a general asset
pricing model extending the results of Cox, Ingersoll, and Ross (1985) as well as
the discrete-time results of Rubinstein (1976) and Lucas (1978) to a multi-agent
Since the CCAPM was first proposed, much effort has been directed at finding
sufficient conditions on the model primitives: the given assets, the agents'
preferences, the agents' consumption endowments, and (in a production econ-
omy) the feasible production sets. Conditions sufficient for the existence of
continuous-time equilibria were shown in Duffie (1986), but the equilibria
demonstrated were not shown to satisfy the additional regularity required for the
CCAPM. In particular, Breeden assumed that all agents choose pointwise interior
consumption rates, in order to characterize asset prices via the first order
conditions of the Bellman equation. Interiority was also assumed by Huang
(1987) in demonstrating a representative agent characterization of equilibrium, an
approach exploited here. The use of dynamic programming and the Bellman
equation, aside from the difficulty it imposes in verifying the existence of interior

Financial support from the National Science Foundation is gratefully acknowledged. We thank
Scott Richard, Kerry Back, Sandy Grossman, Ho-mou Wu, Don Brown, a co-editor, as well as two
referees for comments.

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optima and sufficiently smooth value functions and feedback controls, tends to
emphasize Markovian state information. As we show, the Markov assumption
does not actually play a role in the CCAPM pricing relation.
Our paper unfolds in the following way. Section 2 presents a static analogue of
our approach, showing that the consumption-based and market-portfolio-based
versions of the CAPM are of course the same. Section 3 lays out the model
primitives and the definition of a continuous-time equilibrium. Section 4 states
the regularity conditions on the primitives of an economy that are used to achieve
the representative agent equilibrium pricing model described in Section 5.
The method of proof is roughly as follows. First, we show the existence of an
Arrow-Debreu equilibrium for the static infinite-dimensional economy induced
by complete forward markets at time zero. Under our assumptions, in particular
infinite marginal utility for consumption at zero, we are able to show that every
agent's equilibrium consumption choice is pointwise interior. This marginal
utility condition, however, prevents us from obtaining a uniform properness
condition on preferences, the now standard assumption of Mas-Colell (1986) that
has previously been applied in this setting and elsewhere extended. Instead, we
show that uniform properness can be avoided in this setting.2 Second, following
the method of Duffie and Huang (1985), we implement the Arrow-Debreu
equilibrium as a stochastic continuous-trading equilibrium, using the continuous-
time spanning properties of long-lived securities. Third, we use Huang's (1987)
characterization of the equilibrium spot consumption price process as the marginal
utility of a representative agent's utility for the aggregate endowment process.
In Section 6, the representative-agentpricing formula is used to prove that the
CCAPM applies to the stated equilibrium, and to extend the characterization of
real equilibrium interest rates of Cox, Ingersoll, and Ross (1985) to multi-agent
economies that are not necessarily Markovian.


For comparison with our continuous-time results, consider a static pure

exchange model, with agents defined by endowments e',..., em in the space L of
random variables with finite variance on a given probability space, and by
von Neumann-Morgenstern utility functions ul,..., ur on the real line. There are
also securities in zero net supply with payoffs D = (Do,..., DN) in L, where Do
is a nonzero constant, say Do-1. An equilibriumis a vector S = (So,..., SN) of
security prices such that there exist portfolios 01, . . ., Om in R N+1 summing to

As this paper was being written, we received a manuscript of Araujo and Monteiro (1986), who
have independently obtained a similar abstract existence result. Araujo and Monteiro also obtain
results for the special case of separable utility functions; these results are somewhat different from
ours. More recently, Karatzas, Lakner, Lehoczky, and Shreve (1988) have produced an existence
proof for the special case of separable utility functions, using a much different approach that has
since been simplified by Dana and Pontier (1989). Karatzas, Lakner, Lehoczky, and Shreve (1988)
also provide conditions for uniqueness of complete markets equilibria in our setting.

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zero and solving, for respective i, the problem

max E[ui(e'+O.D)] subjectto S = O.

If, for all i, ui is quadratic and e' E span(D0,..., DN), then the equilibrium
allocation is fully Pareto efficient. (For simplicity, we take it that preferences are
strictly monotone at the equilibrium allocation so as to neglect the usual satiation
problem with quadratic utility.) Thus there are positive weights X1,..., Xm such
that the von Neumann-Morgenstern function ux defined by
ux(c) = max EXiui(ci)
cl + ** +cm C i=l

defines equilibrium security prices by the representative agent pricing formula

sn =E [u'(e)D n], O<-n <, N,
where e = el + * +ee is aggregate consumption. It is easy to show that ux, is
also quadratic, implying that Sn = E[(A + Be)Dn] for some numbers A and B,
which leaves
Sn = KIE(D') + K2cov(e, Dn)

for some other coefficients K1 and K2. If, for any security n, the quantities So,
var(e), and Sn are all nonzero, simple manipulation reduces this asset pricing
model to the classical equation:
E(fen)-r= Pen[E([ffe)-r], (CCAPM)
where n = Dn/Sn is the return on asset n, r = 1/So is the riskless rate of
return, Me= e/( e.- S) is the return on any portfolio3 te whose total dividend is
0 e- D = e (aggregate consumption), and 3en= COV(Me, ?n)/var(te) iS the
"beta" of asset n with respect to the portfolio 0 e paying aggregate consumption.
This is the well known Capital Asset Pricing Model (CAPM), presently in a
slightly unusual guise. In order to see the equivalence between this version of the
CAPM and the traditional (Sharpe (1964)-Lintner (1965)) market-portfolio-based
CAPM, we could equally well view the endowment e' of agent i as the payoff
'y D of an endowed portfolio y' Ej iN?m of securities. In this case, the securities
are held instead in positive total supply M = Yiy' > 0. The portfolio M is the
market portfolio. Of course, since M- D = e, we could take ?ie to be M, so that
the (above) CAPM is the original CAPM

E(Mn) - r = I3Mn[E(qm) - r] (MCAPM)

in which beta coefficients f3Mn = coV (M M,Mn)/var(feM) are measured with

respect to the return 9M = (M- D)/(M- S) on the market portfolio.

3Although the aggregate portfolio of securities is by definition zero, the fact that each agent's
endowment is in the span of the security dividends implies that e is also in their span, implying the
existence of a suitable 9e.

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As one moves from a single-periodto a multi-periodsetting, the CAPM still

applies under suitableconditions,providedone measuresbetas with respect to
current aggregateconsumption,ratherthan the marketportfolio. Suppose,for
example, that the model has multipleperiodsand, for simplicity,that there are
complete security markets.We can let S' denote the marketvalue of the nth
security after its dividend Dn has been realized.The initial price SI of the
securityreflects the total payoff D' + Sn, and is thereforegiven by the pricing
sn = E [u,(e)(Dn + Sn

for a suitable vector A. The consumption-basedCAPM still applies as stated

(CCAPM), once the return on security n is redefined as the total return
9?n = (Dn + S,)/Sn. The analogousequation(MCAPM),substitutingR M every-
where for Me, is not correct,however,unless by some coincidencegM and Me
are perfectlycorrelated,whichcould be truefor example,if endowment"shocks"
are independentlydistributed.Barringextremelyspecial assumptionssuch as
this, it would be a strangequirk indeed if the market-portfolio-based CAPM
applies to a multi-period model.
The representative-agent methodof proof of the CAPM sketchedout above is
extendedto the continuous-timecasein thispaper.By adoptinga continuous-time
formulation,one can drop the quadraticutility assumption,as originallyshown
by Breeden(1979). The resulting"instantaneous"mean and covariancere-inter-
pretationof the aboveCAPMis, for practicalpurposeshowever,no betterthanis
the closeness of utility to quadraticfor consumptionover empiricaltime inter-
vals. By "empirical"time intervals,we mean the time intervalsover which data
are taken and over whichactualconsumersdo not adjusttheirratesof consump-
tion expenditures.As Grossmanand Laroque(1987)point out, transactionscosts
reduce the frequencyof changes in consumptionexpenditures,in some cases
severely, reducing the quality of the relationship:"marginalutility for actual
consumption equals marginalindirect utility for total wealth," the principle
element of the consumption-based CAPM.
As Breeden,Gibbons,and Litzenberger(1986) point out, empiricalresultson
the consumption-basedCAPM are "mixed."The CAPM is neverthelessa good
theoreticalstartingpoint. In fact, for additivelyseparableutility, it's the "only
game in town" as far as demonstratedequilibriummodels of securityreturns.
(For an examplewith the more generalrecursiveutility formulation,see Epstein
and Zin (1989).) The market-basedmulti-periodCAPMdoes not generallyhold,
as shown by Merton(1973),and as pointedout againabove.It is far from dead,
however,as a practicalmodel.Chamberlain(1988)recentlydirectednew theoreti-
cal attention toward an intertemporalmarket-basedCAPM, based on an as-
sumed equilibriumsatisfyingcertainproperties.The key propertyChamberlain
assumed, aside from mathematicalregularityconditions, is that the marginal
utility for aggregateconsumptionof the representativeagent,an analogueto the
random variable u'(e), is a (measurable)function of some one-dimensional
Brownianmotion. In the generalsettingof multi-dimensionalBrownianmotion,

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one cannot claim that uk(e) dependsonly on a singleBrownianmotion,although

that may happen in specialcases. For example,if uncertaintyis generatedonly
by a one-dimensionalBrownianmotion, then the market-basedCAPM is in
principle correct, but tautologicallyso, since everv Ito process is "instanta-
neously"perfectly correlatedwith every other Ito process in that case. That is,
the "9-portfolioCAPM"is correctfor any portfolio 0, with a one-dimensional
Browniansourceof uncertainty.


We quicklylay out the model primitivesand the definitionof an equilibrium.

Backgroundinformationcan be found, for example,in Duffleand Huang(1985)
and Duffie (1986).
We work with a finite time interval[0, T], a probabilityspace (2, Y, P) on
which is defined a standardBrownianmotion B in RK, and the augmented
filtrationF = { F: t E [0, TJ} of sub-tribes(a-algebras)of Y naturallygenerated
by B. Unless otherwiseindicated,all probabilisticstatementsare made,suippress-
ing "almost surely,"relativeto the filteredprobabilityspace (Q, J, F, P), our
basic model for informationand beliefs. At each date t in [0, T], there are
markets for securitiesand a single consumptioncommodity.Our consumption
space is the vectorspace L of square-integrable predictablestochasticprocesses.4
Informally,if x E L is a consumptionprocess,then at any time t, the consump-
tion rate xt is based only on information available up to that time, and
E(f 0Tx2dt) < 00.
Each of the m agentsin this (pureexchange)economyis representedby a pair
(U , e'), where Ui is a utility function on the (usual) positive cone L+ of
consumptionprocessesand e1 e L, is an endowmentprocess,for i e {1,..., m
The remainingprimitivesare financialsecuritiesin zero net supply paying
dividendsin units of account,just as in Arrow's(1953) originalpaperon general
equilibriumwith securities.We designateone of these securitiesas a numeraire
and for simplicitytake it to be a pure discountbond redeemableat the terminal
date T for one unit of account.This is not a monetarymodel, but until we later
convertprices to the consumptionnumerairein orderto characterizereal security
prices, it would be equivalentto treat the numerairesecurityas inside money.
Since the total supplyof each securityis zero,in equilibrium,no agentis required
to "carryaway" any of the securitiesafter tradingis complete at the terminal
The cumulativedividendsof the numerairesecurityare describedby the process
DO defined by Dto=0, t < T, and DO= 1. The remainingsecurities,say N in
number,are representedby cumulativedividendprocessesD1,..., DN, where,for
each security n, Dn is an Ito process such that DT has finite variance.By Ito

4 Formally, L = L2(S2X [0, T], 9, v), where S9 is the predictable tribe on Q2x [0, T] (that gener-
ated by the left-continuous adapted processes) and v is the product of P and Lebesgue measure on
[0,T] restricted to .9.

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process,we mean that D satisfiesa stochasticdifferentialrepresentation

dDt = MD(t) dt + aD(t) dBt,
whereAD is an N-dimensionalpredictableprocessand a is an N x K-dimensional
predictableprocess. (Thereis no Markovianassumption;,uD(t) and GD(t) may
depend on the entire past historyof the economy.)The entire economy is thus
describedby a collection
(1) = ((2,Y,F, P),(Ui,ei), D), iE {1,..., m),
whereD = (D?,..., DN) is the (N + l)-dimensionalcumulativedividendprocess.
Agents take as given a spot consumptionpriceprocessp e L and an (N + 1)-
dimensional Ito security price5 process S = (S,..., SN). The gain process G =
(S + D) is thereforean Ito process, and we can write dGt= ,t dt?a + dB, for
some yt and a. This allows one to define cumulativegains from trade for a
predictableportfolio process 9 = (90,..., ON). A fixed portfolio 0E9 R+N? 1, for
example, generatesfrom some time T until some time s > T the interim gain
Jf'sdGt= 9 - [(Ds - D) + (Ss - S7)], the obvious sum of dividend gain and capital
gain. For regularity,we demandthat a portfolioprocess 9 be square-integrable,
O * t dt be well-defined,
meaning E( JoT9t aiAcT9 dt) < ox, and that the integral fOt
meaning JoT1It9, Iy dt < oc almost surely.These regularityconditionsregardinga
portfolio process 9 imply that the gain-from-trade integral J9, dG, = E oJfOtndGtn
is well-defined, and further that Ja, dG, is a martingale whenever G is a
martingale.The vectorspace & of all portfolioprocessesincludesstrategiesthat
vary a portfolio continuallyin time, a tradingmodel inauguratedby Merton
A budget-feasibleplan for agent i is a pair (x, 9) E L+ x 9 such that, for any
time t,

(2) at*St= f'9sdGs + p(e -xs) ds,

and such that OT =0. Relation (2) is merely the accounting restrictionthat
current portfolio wealth must be generatedonly from trading gains and net
consumptionpurchases.The terminalrestriction9T= 0 (no terminaldebts)is the
essential budget constraint.A budget feasibleplan (x, 9) for agent i is optimal
for agent i if thereis no budgetfeasibleplan (x', 9') such that U'(x') > U'(x).
An equilibrium for an economy 9' is a collection ((S, p), (xi, 91),... , (xm,l M))
such that, given the securityprice process S and the consumptionspot price
process p, for each agent i the plan (x', Oi) is optimal, and markets clear:
Eix'-e1= 0 and E9iO=O. Although Duffie (1986) shows conditions on an
economy sufficientfor the existenceof such an equilibrium,those conditionsare
not sufficientfor the asset pricingmodelsof interesthere. AppendixC discusses
the issue of nominal versus real (consumptionnumeraire)primitivedividend

S We take the convention of cum dividend security prices, so that ST, for example, is the market
value at time T of any lump sum dividends paid at time T.

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processes, showing that we can accommodate the latter by more complicated

arguments. Just as in Arrow's (1953) paper, however, it is much easier to describe
spanning assumptions for purely nominal securities. Duffie (1986) indicates how
the assumption of Ito dividend processes can be relaxed.


We consider the following conditions on an economy.

(A.1) For each agent i, Ul is representedby a smoothfunction u1: R +X[0, T] R

in the form

(3) UI(x) = E [jui(xt t) dt]

where, for each t in [0, T], the function ui(, t): R +- R

o is strictly concave,
increasing, with the first derivative on (0, oc) denoted uic(, t) satisfying
limk I IO uic(k, t) = +0.

By smooth, we mean that for any e > 0 sufficiently small, the restriction of u1
to (?,e + e) X [0, T] is C', where e is the essential supremum of the aggregate
consumption process (which might be + ox). A C? function is one with an
extension to an open set having continuous derivatives of any order.6 This can be
relaxed.7 A frequently used example satisfying (A.1) is the function u(k, t) =
exp (- yt)ka for some y > 0 and a E (0,1). Lehoczky and Shreve (1986) have a
closed form solution for this case, provided agents have the same utility function.

(A.2) The aggregate endowmentprocess e = Em 1e' is an Ito process, bounded

away from zero, where the stochastic differential representation det = Mue(t)dt +
ae(t) dBt is such that E(JOTae(t) *ae(t) dt) < ox.

Again, there is no Markov assumption here; ye(t) and ae(t) can depend
generally on the entire history of the economy up to time t. The assumption that
e is bounded away from zero, however, is restrictive (see Araujo and Monteiro
(1987)), and is used to ensure that spot prices for consumption are bounded,
given the "infinite marginal utility at zero" assumption in (A.1). With this
boundary condition on preferences, aggregate consumption might naturally be
bounded away from zero in a production economy with suitable technological
possibilities. We have limited ourselves to pure exchange economies for simplic-
ity; in principle, our results should extend to production economies under natural
regularity conditions.
The following is a spanning assumption; because of this assumption, in
equilibrium any conceivable consumption plan can be financed by some portfolio

6We later use the fact that continuity of the derivatives and concavity of u&(, t) imply uniform
convergence of the limit indicated in (A.1).
7Working through our proofs shows that a C3 assumption on ui would more than suffice.

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process at some initial cost, and any conceivable additional security is redundant.

(A.3) The martingalesM1,..., MN defined by

(4) Mtn= E [ DTIt ] t E-[O, T]E

form a martingale generator.

Assumption (A.3) means that any martingale Y can be represented in the form
y,= Y?+ ftOndM

for predictable processes 91,..., ON stochastically integrable with respect to

M1,..., MN. Since any martingale such as M = (M1,..., MN) has a stochastic
differential representation of the form dMt= qgtdBt, where g is a N X K matrix-
valued process, for (A.3) it is necessary and sufficient that the rank of g is K
almost everywhere (implying that N > K). For a trivial example, let Dn = B ,
1 < n < K. The "log-normal" case Dtn= exp (Btn), 1 < n < N can also be shown to
satisfy (A.3).
Assumption (A.3) can in fact be weakened to an assumption that, under
substitution of some probability measure Q uniformly equivalent8 to P, the
martingales defined by (4) form a martingale generator under Q. (See Appendix
C, or Duffie (1986) for further details.)


A representative agent for a given equilibrium ((S, p), (xi, 01),... (xmM))
for & is a single agent (Ux, e) having: (a) the same total endowment e of 9, (b) a
utility function U; defined by
(5) Ux(x) = sup FXiUi(x') subject to Fxi < x,
(x. xm) L+

for some coefficient vector X E t +, and (c) the same equilibrium price processes
(S, p) (in a no-trade equilibrium) for the single-agent economy &<=
((,Q, _F, F, P), (Ux, e), D).
Under preference assumption (A.1), we know that if Ux is defined by (5), then

Ux(x) = E[f U(Xt, t) dtj


ux(k, t)= sup EXiui(ki,t) subjectto Eki<k.

(k1,...,km)lRm i i

8A probability measure Q is uniformly equivalent to P if the Radon-Nikodym derivative dQ/dP

is bounded and bounded away from zero.

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From this fact, it is easy to see that Ux also satisfies preference assumption (A.1),
using the implicit function theorem to demonstrate smoothness.
Huang (1987) demonstrated a representative agent in this sense for a given
Markovian continuous-time equilibrium satisfying certain regularity conditions.
Our first result establishes the existence of an equilibrium with the required
regularity conditions, and the obvious analogue of the Lucas (1978) "stochastic
Euler equation." Later, we extend the result to obtain a more general pricing
model. We let ux,(-, t) denote the first derivative of u&(*, t). The following result
provides for the existence of an equilibrium ((S, p), (xi, 91), ... ,(xm, tm)) for e
and characterizes the real (consumption-numeraire) security price process St=
St/pt in terms of the representative agent's marginal utility ux,(e ' t) for aggre-
gate consumption and in terms of the real dividend process D, defined by

dDt = pt- 1 dDt.

THEOREM 1: Under conditions (A.1)-(A.3) on the economy , there exists an

ejuilibrium with a representativeagent (Ux, e) such that, for any time t, the vector
St of real security prices satisfies the representativeagent pricing formula

(6) [,uxc(es, s dDsJt ttE- [0, T)

Moreover, for any agent i with e1 # 0, the equilibriumconsumptionprocess xi is
bounded away from zero.

PROOF: Step A-The Arrow-DebreuEquilibrium: By Appendix Theorem A4,

there exists an Arrow-Debreu equilibrium (xl,..., xm; ?T) for the complete mar-
kets economy (Ui, ei), where the price functional 7r: L -0 R is represented by
some peL + in the form

T(x) =E[fT dt] x EL,

with p bounded and xi bounded away from zero for any i with e' # 0. This is an
extension of the work of Mas-Colell (1986), Yannelis and Zame (1986), and
Zame (1988) to the case of preferences that are not uniformly proper. (See
footnote 1 for related static equilibrium existence results.) Steps A and B of the
proof use only Assumption (A.1) plus the assumption from (A.2) that e is
bounded away from zero.
Step B-Representative Agent Pricing: We recall the representative agent
construction of Huang (1987). Briefly summarizing, by the saddle point theorem,
there exist nonnegative constants X1,...,Am (with Xi= 0 if and only if e'= 0)
such that the aggregate consumption process e solves the representative agent

max LE f u(xt, t) dt] subject to 7T(x) < XT(e).

Then, by the first order conditions at the solution e to this problem, there exists a

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Lagrange multiplier y > 0 such that ux,(e,, t) = -yp almost everywhere. For the
details, see Huang (1987). Without loss of generality, y = 1.
Step C-Dynamic TradingImplementation: In roughly the sense of Duffie and
Huang (1985), the static equilibrium ((x'), 7T)can be implemented by security
trading strategies (01,..., Om) to form a stochastic equilibrium ((S, p), (x', 0'))
with the nominal security pricing formula St = E[DT - D, JI.
To outline the basic parts of this implementation procedure, pick any agent i
and let Y be the martingale defined by

Yt= E( PS(x -eDds J)-

The gain process G = S + D is a martingale generator by the spanning assump-
tion (A.3). By the definition of a martingale generator, there exist predictable
processes T1 TK such that

Y=YO+E |<psdGsk t[09 T].

Let ?9 be the predictable process defined by

= Y-I
:v (xi -ei) ds-E tS
O ~~~k=i
An exercise shows that the trading strategy 0i = (qP, 19 T... K) forms a budget
feasible plan (x', O1) for agent i. A simple proof by contradiction suffices to show
that (x', 01) is moreover optimal for agent i. We can choose such a plan for each
of the m agents. Now, suppose we replace the trading strategy 0Omso chosen for
agent number m with the new trading strategy 0 m =- i l7jll. This implies
clearing in the security markets. Consumption market clearing in the given
equilibrium for the static economy (as well as the linearity of integration) then
implies that (Om,cm) is budget feasible for agent m. Since (Om,cm) is optimal for
m, it follows that (OM,cm) is also optimal for m. Thus we have shown the
existence of an equilibrium in the stochastic economy.
Step D-Renormalization of Prices: Using the definition of the consumption-
numeraire prices and dividends, (S, D), given before the statement of the
Theorem, and the fact that Pt = uxj(et, t), we recover the security pricing formula
(6). Q.E.D.

By taking any two times t and T >, t, we can substitute the expression for ST
given by (6) into the corresponding equation for St to verify the analogue to
Lucas' (1978) "stochastic Euler equation,"
A 1 rT]
St = E uxc(es s) dDs+ uxj(e, T)ST J ]
Ucux(t, t) LJ c]
It is convenient (and useful for purposes of generality) to consider more
general forms of security dividends. To begin, we consider a cumulative real

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dividend process in the form of an Ito process Y having a final value YT with
finite variance. By definition, Y solves a stochastic differential equation of the

dYt= ,uy(T) dt + or(t) dBt,

where iyyand ay are (possibly path-dependent) processes. (We take iyyand ay
to be predictable processes, real-valued and R K-valued,respectively.)
Introducing the new security Y into the model changes nothing in equilibrium,
given the spanning assumption (A.3), provided the new security is priced consis-
tently with equation (6), that is, provided the current real price of the security is
1 [T]
Sty u(e E Iuxc(es s) dYs F ]
uxc(et,t) LJct
(Since the security is redundant, any other price process allows arbitrage.)
Likewise, any additional security paying a terminal lump sum dividend 8 is also
redundant, and at time t has the equilibrium real price E[uxc(eT, T)S iFt]/
uxc(et, t). (We only assume that 8 is YT-measurable and of finite variance.) By
additivity, the equilibrium price process sy8 for a security promising the Ito
dividend process Y as well as a terminal dividend 8 is therefore defined by
1 rT1
(7) StY8= F(et [I[ uxc(es, S) dYS+ uxC(eT,T)> JFt j t [0, T].

This is formalized by the following corollary to Theorem 1 for the augmented

economy 9`6 defined by the original economy &= ((Q2,Y, F, P), (U', e1), D))
and the additional security paying the real dividend process Y as well as the
terminal real dividend S. (The result extends to the augmentation of any number
of additional securities.)

COROLLARY: Further to the statement of Theorem 1, for any cumulative (Ito)

dividendprocess Y of finite variance and any terminal lump sum dividend 8 of finite
variance, the augmented economy 9`6 has an equilibriumwith the same consump-
tion allocation and the same representativeagent in which the real price process S Y
for the security (Y, 8) satisfies equation (7).

Intermediate lump sum dividends paid at (possibly random stopping) times

can also be priced in the obvious way, as shown in Duffie (1986).


We will now characterize the equilibrium excess rate of return on any security
paying a cumulative (Ito) real dividend process Y plus some lump sum terminal
dividend S. For notational convenience, we let V denote the equilibrium real
price process S' of this security, as defined by (7).

All use subject to JSTOR Terms and Conditions

Consider the process Z defined by Z, = Jo4p,dY, + p,V,. From (7), it is easy to

check that Z is a martingale.Since V is an Ito process,we can alwayswrite
dVt= ,uv(t) dt + av(t) dBt,
for some predictableprocess,i v and RK.valued predictableprocessaVIAlso, by
Ito's Lemma(as statedin AppendixB),
dZt= [ptAy(t) +p,.v(t) + Vttp(t) + uxcc(et,t)Ue(t) - av(t)] dt
+ aZ(t) dBt,
for some process oz that we need not calculate.Since Z is a martingale,for
almost every time t,
Pt [ (t) + 1v(t)I + Vttp(t) + uxcc(et, t)Ue(t)
- ov(t) = 0.
Dividing through by pt = uXJ(et,t), which is everywhere nonzero, we have the
basic rate of returnrelationship

(8) Av (t ) + tt(t) - rtVt= ; a v(t) a(t )

where rt =-ip(t)/pt. For Vt' 0, we can define the total real rate of expected
returnon this securityat time t as
_ [AV(t) + AY(t)]
the expected real rate of capital plus dividendgain divided by currentmarket
value. Letting aq(t) = av(t)/V(t), it is common to treat q,(t) -a (t) as the
"instantaneousvarianceof return"of the security.(After completingthe state-
ment of the CCAPM, however, we make a critical remark concerning the
generalityof this definition.)We now have the more familiarCCAPMrelation

(9) St - rt= -ux(et' t) asq(t) ae(t),

taking the common interpretation of aq(t) *ae(T) as the instantaneousconditional
covariancebetweenincrementsof aggregateconsumptionprocesse and the total
real return on the given security.The securityin question is riskless (in real
terms) if au(t) 0. Equation(9) then gives us a characterizationof the riskless
rate of return, rt= - Ap(t)/p,. Since Pt = uxc(et, t), the riskless rate of return or
equivalently,the real equilibriumshort terminterestrate, is minus the expected
rate of growthof marginalutilityfor consumptionof the representativeagent.In
a more restrictivesettingof a single agent and a Markovianstate process,Cox,
Ingersoll, and Ross (1985) have previouslygiven this characterizationof the
More formally,we have the followingversionof the CCAPM.

PROPOSITION1 (CCAPM): There exist equilibriafor economies satisfying As-

sumptions (A.1)-(A.3). In equilibrium, at almost every time t, the riskless real rate

All use subject to JSTOR Terms and Conditions

of return is r= - p (t)/p, (minus the expected rate of change of marginal utility

for consumptionof the representativeagent), and the expected real rate of return t
of any security satisfies (9), provided the market value of the security is not zero.

To describe a* a as the "instantaneous variance of the return of the

security" is quite misleading unless the security Y, defined by dYt= iuy(t) dt +
ay(t) dBt, has ay(t) = 0. That is, if the security's dividends are "instantaneously
riskless," as is said, then the nomenclature is appropriate. Otherwise, a actually
corresponds to the "instantaneous variance" of capital returns only, and does not
include the effects of the "local risk" of dividend returns.
One can also give the usual "beta" interpretation of relation (9) as follows.
Consider any security whose equilibrium real price process Ve has a diffusion IVe
such that aVe(t)/Ve= ae(t) for all t. One can always design such a security by
our spanning assumption (A.3). Since the excess expected real rate of return of
this security, denoted Me, satisfies the CCAPM (9), we have
XCet t)
(10) ~ gt er _
rt U=c(et ) (ye(t) *(Te(t).
Combining (9) and (10), we have the traditional beta relationship
(11) 9P-t=3(t)(qe-rt)

ae(t) * e(t)9

assuming the denominator is nonzero.

Relationship (11) is not equivalent to (9), since (11) is true even when (9) is
adjusted by replacing the representative agent risk aversion measure
- uxcc(et, t)/uXJ(et t) with any well-defined process. That is, relation (11) states
merely that excess expected returns are proportional to consumption betas, while
relation (9) also specifies the constant of proportionality. Breeden's original work
suggests that (11) will in fact hold without spanning in an appropriately regular
equilibrium, using a suitable definition of Me as the excess expected return of a
particular portfolio with "returns most highly correlated with aggregate con-
sumption increments." Existence of nontrivial continuous-time equilibria with
incomplete markets is an open issue. Certainly, spanning is not an issue in a
single-agent economy, and (11) can easily be shown to hold in Breeden's sense in
a single-agent economy, Markov or not, following the calculations made here,
since the equilibrium allocation in a single-agent economy is Pareto optimal.
Aside from the single-agent case, or except by a strange coincidence, incomplete
markets negates both Pareto optimality and a representative agent.


We remark on several extensions of the theory in this paper. As pointed out by

Breeden (1979), the CCAPM extends, with limitations, to the case of multiple

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commodities. Back (1988) has giveii a partial extension to more general informa-
tion structures than that generated by Brownian motion. Of course, our main
result (Theorem 1) and the representative agent asset pricing model (6) apply to
an arbitrary information filtration satisfying the usual conditions, provided the
integral in equation (6) is well-defined. It is only when we derive the CCAPM
itself that we turn to the "instantaneous covariance" characterization of asset
returns, which depends on a Brownian information structure. Our abstract
existence results in Appendix A also permit an extension of Theorem 1 to the
case of multiple commodities and an additive state-dependent utility function ui:
Q2x [0, T] x R 1- R, subject to technical conditions. Of course, the CCAPM is
not true with state-dependent utility.
Finally, we remark that our results meet Huang's (1987) sufficient conditions
for Markovian equilibrium. That is, if e and D are well behaved functions of a
Markov state process X, then the equilibrium (real) security prices defined by (6)
are also measurable functions of the state process X.

Graduate School of Business, Stanford University,Stanford, CA 94305, U.S.A.

Department of Mathematics, State Universityof New York, Buffalo, NY, 14214,

ManuscriptreceivedMay, 1987; final revision receivedMarch, 1989.


This Appendix presents the required existence result for the static Arrow-Debreu economy. The
precise result we require is rather special, but the underlying result is rather general. Since this
underlying result seems potentially useful in other contexts, we digress slightly to present it first.

AbstractArrow-DebreuExistence Result
Let K be a normed vector lattice with positive cone K+ (see Schaefer (1974)). We consider an
Arrow-Debreu (pure exchange) economy 9'= (?, e') in K, with m agents described by preference
relations > i on K+ and initial endowments e' E K+. Throughout, we make the usual assumptions
on preferences: each >- i is reflexive, transitive, complete, convex, continuous, and strictly monotone.
We say that a (Hausdorfl) topology T on K is compatible if it is weaker than the norm topology
and if all order intervals [0, x] = {y e K: 0 < y < x } are r-compact. We say that the preference
relation >- satisfies the forward cone condition at x E K+ if:
(FC): There isaveK+, an E> 0 anda p>0suchthatx+Xv-z>-xwheneverx+Xv-zeK+,
O< X < p, and llzll < Xe.
The forward cone condition was introduced by Yannelis and Zame (1986), and is a variant of the
backward cone condition ("properness") introduced by Mas-Colell (1986). The validity of the
uniform version of the forward cone condition (that is, that (FC) holds at each x E K+, with v, E,p
fixed) is equivalent to uniform properness, which was used by Mas-Colell to obtain the existence of
competitive equilibria. However, the pointwise versions of the forward cone condition (which we use)
and of properness are incomparable.
An allocation (xi,. . . xm) E (K+)m is feasible for d' if E, (e' - x') = 0. A competitiveequilibrium
is a feasible allocation (xl,. . ., xm) and a linear functional v on K such that, for all i, v - xi < T*ei
and x'>-1yi for any y .E K+ with qr.yi < i7 * ei. A feasible allocation (xl, ... , xm) is in the core

All use subject to JSTOR Terms and Conditions

if there exists no other allocation (yl,..., y i) and nonempty subset W of agents such that
? (e' - y') = O and y' I xi for all i c- -.
Our abstract existence result is:

THEOREM Al: Let K be a normed lattice, let r be a compatible topology on K, and let 4) be an
Arrow-Debreu economy in K. Assume that: (a) each of the preference relations >-, is T-upper
semicontinuous; (b) for each core allocation (xi,... xIn) for 4), each >-, satisfies the forward cone
condition at x'; and (c) e = 2,e' is strictlypositive. Then 8has a competitiveequilibrium(x,...,xxm; r),
where i7 is a positive continuouslinearfunctional on K.
To prove Theorem Al, we first consider a restricted economy 4. Let K(e) be the order ideal

K(e) = {xe K: 3s +R: Ixi ?se},

where Ix =x+ + x-, and define a norm 11 Ile on K(e) by

lixile=inf{s>O: IxisIse}.

It is easily checked that, equipped with this norm, K(e) is a normed vector lattice. The economy 4) is
simply the restriction of 8) to K(e). That is, 4) has the same agents (1. m}, with the same
endowments el,. . ., em, and with the preferences obtained by restricting a, to K(e).
The following lemma and its proof are slight variations of those in Zame (1987) as well as
Aliprantis, Brown, and Burkinshaw (1987a). The general idea of working on order ideals of the
consumption space, allowing one to naturally extend Bewley's (1972) approach, can also be found in
Brown (1983) and in Aliprantis, Brown, and Burkinshaw (1987b).

LEMMAA2: The economy 4 has a competitive equilibrium (xl..., xm; ii), where X7 is a positive
linear functional on K(e) which is continuouswith respect to the norm 11 Ile.-

PROOF:Let f denote the restriction of T to K(e). The feasible consumption sets for &eare the
order ideal [0, e], which is r-compact (by assumption). Moreover, the endowment e lies in the interior
of the positive cone of K(e). It follows9 from Theorem 2 of Zame (1986) that 4) has a quasi-equi-
librium'0 (xl,..., x-; ST), where -6 is continuous with respect to the norm 11 lie. Strict monotonicity
of preferences implies as usual that 'i is positive and that (xi,.. I xm; *F)is an equilibrium. Q.E.D.

LEMMAA3: If (xi,... I x M;-7) is an equilibriumfor 66 and if each of the preference relations >-
satisfies the forward cone conditionat x', then 6 is continuouswith respect to the original norm II on

PROOF:11 Let v,,Ii, and pi be as in the forward cone condition for >- i at x'. At the cost of
replacing ?, by E,/2, we may replace v, with any vector v1'E K+ such that lIv,- vI < E,/2. Since
K(e) is dense in K, we may without loss of generality assume that vi E K(e) for each i. Set v =E,v,
and E = min{e1..,}. We claim that

I,ff -YI < -IIYII,

IF V Y E- K(e)-

Note first that (since 'i is positive) it suffices to establish this for each y c K(e)+ with IlYll< 1.
Suppose that i *y > ST *V/E. For X > 0 small enough, Xy < e, so e-Xy E K(e)+. Since e =?,e' =
E,x', the Riesz Decomposition Property (Schaefer (1974)) allows us to find vectors z, E K(e) such

9 Theorem2 of Zame (1986)assumesthat each of the endowmentse' lies in the interiorof the
consumptionsets, but this assumptionis usedonly to obtainan equilibriumfroma quasi-equilibrium
(no monotonicityassumptionis used there).
10A quasi-equilibrium is the same as an equilibrium,exceptfor the substitutionof the condition
U(x') > U(x') * * x x' in place of the usualoptimalitycondition.
11This is the argumentof the PriceLemmaof Yannelisand Zame(1986).

All use subject to JSTOR Terms and Conditions

that 0z, 6 x' and AAy=izi. Then

e+ Av - cAy= (xi+ Avj -z,).

SinceE2zi= EAy,it followsthat Ilzill6 E,jjAyjj for each i, and hence(by the forwardcone condition)
that x' + Av, -ziz>- x' for each i. On the other hand, iF*(Av - EAy) < O so . - (x' + As, - z1) < I*. xi
for some i. This contradictsthe equilibriumpropertiesof (x',..., xm;if), so we conclude that
f *y 6 (i - v/E) jyjjfor each y E K(e)+, andhencethat * is continuouswithrespectto the 11 HI-norm
on K(e), as asserted. Q.E.D.

PROOF(of TheoremAl): By LemmaA2, the economyg has an equilibrium(xi,... xm;if). The

equilibriumallocation(xi,... xm) is in the coreof o; since all feasibleallocationsfor of are actually
in K(e)", the allocation(xi,..., xm) is in the coreof 4. SinceK(e) is densein K, the continuityof
Xi(LemmaA3) implies that so has a (unique)extensionto a positive,continuouslinearfunctionaliT
on K. Continuityof preferencesimpliesthat(xi,. . . xm;iT)is an equilibriumfor 4. Q.E.D.
An Applicationto the L2 Case with Additive-SeparablePreferences
We turn now to the preciseresult required.From now on, the commodityspace L = L2(s2 X
[0, T], .9, v). Let 4= (>- ,, e') denotethe corresponding completemarketseconomyon
L, where >-, is represented by a utility function U'.

THEOREMA4: Under condition (A.1) of Section 3 and the assumption that the aggregate endowment
process e is bounded away from zero, 4 has a competitive equilibrium (xi.... xM; v), where 'r is
representedby p E L+ as in relation (6) withp boundedand, for all i, if e' *0, then xi is boundedaway
from zero.

PROOF:Let K denote L equippedwith the norm 11 givenby

lixil = E(j )xt dt).

As such, K is a normedvectorlattice.It is easily checkedthat each of the preferencerelationsis

continuousin the 11 1l-topology. Let Tdenotethe weaktopologyon K withrespectto the topological
dual K* of L. This topologyT is compatible.'2Moreover,since norm-closedconvexsets are weakly
closed, the preferencerelations >- i are T-uppersemicontinuous.
Fix a measurable(predictable)subset A of Q2x [0, T] and a real numbera > 0. If y E K+ and
y > a on A, then togetherwith the MeanValueTheoremour assumptionsimply that

(13) U'(y) - U'(y - z) < E( Tz,dt) sup ui(a2t)

Ote[0, T]

wheneverz E K+ is any vectorin K+ supportedon A andboundedaboveby a/2. Similarly,for any

set H E. 9, h > 0, and y E K+ with y S h on H, we obtain
(14) U'(y+X1A)- Ui(y)1>Av(H) inf u,c(3h/2,t)
E[O, T ic

whenever0 < A< h. Thereis no loss of generalityin assumingthateache' is nonzero,andhencethat

U'(ei) > Ui(0) for each i. Suppose (xl,..., xm) is a core allocation for S. Since U'(x')> U'(e') >
U'(0), thereis a set A, c.9 with v(Ai) > 0 anda numbera, > 0 with x' > a, on Ai. If some xi is not
boundedaway fromzero,we can findfor each h > 0 some He-9 such that v(H) > 0 and xi < h on
H. Since e is boundedaway fromzero, say e > c > 0 for some c, we can assumewithoutloss that
xi > c/m on H for some agent 1. Applying(13) and (14) and our assumptionthat u,c(k, t) -+ oo as
k -- 0, we conclude that agents j and I could exchangeappropriatemultiplesof the commodity
bundles1H and 1A to effecta Paretoimprovement providedh is smallenough.Sincethis wouldbe a
contradiction,we concludethateach xi is indeedboundedawayfromzero.
On order intervals,this topologycoincideswith the weak topologyon L with respectto L*;
Alaoglu'sTheoremimpliesthatorderintervalsare compactin this topology(Schaefer(1974)).

All use subject to JSTOR Terms and Conditions
We now assertthat the forwardconeconditionshold at the points x. Xm.Sincethe x"s areall
boundedbelow away fromzero and boundedaboveon some set C E 9 with v(C) > 0, this follows
immediatelyfrom (13) and (14) (takingvi = lc for each i). We now concludefromTheoremAl that
d' has an equilibrium(xi I., xm; f7). Since(xl,... Ixm) is a coreallocation,xi is boundedawayfrom
zero for each i. Moreover,since v is 11-1ll-continuous,
the Rieszrepresentation theoremimpliesthat ST
is representedby a positiveboundedfunctionp; since v is a finitemeasure,p is in L. Q.E.D.

We have used Ito's Lemmarepeatedly,and it maybe best to stateit formallyfor the recordin the
setting of Ito processes.If dXt= ,ux(t) dt + ux(t) dBt is an Rn-valuedIto processfor some n, and
f: R"nx [0, T] -D R is C2 (whenextendedto an open set), then the processYdefinedby Yt=f(Xt, t)
is also an Ito process with stochastic differential dt + ay(t) dBt defined by ay(t)=
fx(Xt, t)ax(t) and

ALy(t) =fx(X , t)*x (t) +ft(Xt, t) + trace [x(t)Tfxx(Xt, t)Ox(t)],

wherethe vectorfx( Xt, t) and matrixfxx( Xt, t) denote the firstand secondpartialderivativesof f
with respectto the X arguments.(The same formulaappliesunderweakerconditions.)We actually
use two specialcases of this resultin the paper.One is the case of n = 1, Xt = et, and Yt=f ( Xt, t) =
U.c(et, t). In this case, ay(t) = uxcc(et, t)oe(t). The secondapplicationof Ito'sLemmain the body of
the paper is the case of n = 2, XtM)-P,,pt,Y, = and =f(Xt, t) = Xt1lXt2). In the latter case,
applicationof Ito's Lemmaleads to
Ay (t) =PtAv (t) + VtJp (t) + UV(t) *op(t).
Details and extensionscan be found,for example,in Lipsterand Shiryayev(1977).

Just as with Arrow's(1953)modelof securitymarkets,we havefoundit convenientin this paperto
define the primitive,exogenouslygiven,securitydividendsin (nominal)unit of account,not in (real)
consumptionnumeraireterms.This allows an easy developmentof spanningin continuous-time,
based on primitiveassumptionson the nominaldividendprocess D. In a generalmulti-commodity
model, thereis little alternative,as shownby Hart (1975),since equilibrianeed not exist in general.
(AlthoughHart workedin a discrete-timesetting, the fact that the span of marketsmay collapse
discontinuouslyat certainendogenousspot priceprocessesdoes not disappearin continuous-time; it
only becomesmoredifficultto dealwith.)In the single-commodity settingof thispaper,however,one
can definespanningassumptionsdirectlyon a real riskysecuritydividendprocessy = (Yy.yN),
provided there is also a nominalnumerairesecurityDO as definedin Section 3. In this case, the
primitives of an economy are J%=((Q, -F, F, P),(Ub, e'),(Y, D?)).
Suppose,for N > K, that Y is an RN_valuedIto processwith stochasticdifferentialrepresentation
dYt = uy (t) dt +ay (t) dB,
and that Mt= Jjtry(s) dBs is a martingalegenerator(as definedin (A.3)). For this, it is basically
enoughthat the essentialinfimumof the rankof ay is maximal,and thereforeequalto the dimension
K of the Brownianmotion B. As an alternativeto (A.3), this is in principleenoughto demonstrate
the existenceof an equilibriumsatisfyingall of the resultsof the paper.To guaranteethis, we need
only completethe proof of Step(C) of Theorem1, as follows.
We take static equilibrium(xl,..., xm; in) as givenby AppendixA, TheoremA4, whereif has the
representationff. c = E(fJTptcdt), for p boundedand boundedaway fromzero.The corresponding
nominal dividendprocess D' is definedby dDt'=pt dYt'.By Girsanov'sTheorem(undertechnical
regularityconditionson .&y and ay(t) given in Lipsterand Shiryayev(1977)), thereexists a new
probabilitymeasureQ uniformlyequivalentto P and a Brownianmotion B relativeto Q under
which dDt=pAay(t)dBt. Since p is bounded above and below away from zero, the spanning
assumption(A.3) is thereforesatisfiedunderthe new measureQ. Let S, = EQ[DT - Dt[JFt definethe
nominalsecuritypriceprocesses,whereEQdenotesexpectationunderQ. In orderto definea suitable

All use subject to JSTOR Terms and Conditions

new consumptionspot priceprocesspl, firstdefinethe densityprocess

Zt= [dP t ], t[O,T].

Now let Pt=Pt/Zt for all t. By Lemma4.1 and Proposition4.1 of Duffie(1986),thereexist trading
strategies (61. f) such that ((S,p),(xl,61). (xm,Gm)) is an equilibrium for e'. In order to
recoverthe pricingformula(6) on whichthe rest of the paperis based,we use the fact that,for any

Combiningthis rule with the argumentsused in the proof of Lemma4.1 of Duffie(1986),it is then
easily shown that the realpriceprocessS definedby St= t/At satisfiesthe basic representative
pricingequation(6), and the papercontinuesas before.
This Appendixshows that,in this specialsetting,one can actuallyprovideconditionsdirectlyon
real primitivesecuritydividendsthat generatespanningin equilibrium.Aside frombeing somewhat
more complicatedand requiringadditionaltechnicalregularityconditions,this yields the same asset
pricingresultsstatedin the body of the paper.

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