A continuous time asset pricing model is developed in the context of a pure exchange economy with a single perishable consumption good, a representative agent and complete markets. The good serves as the numeraire. The economy has a finite horizon Œ0; T . The uncertainty is carried by ad-dimensional Brownian motion processW. Brownian increments represent economic shocks. There arek state variablesY. Aggregate consumptionC, dividends D and state variablesY follow diffusion processes.
3.2.1 The Financial Market
The financial market has ds risky assets (stocks) and 1 locally riskless asset (a money market account). Stocks are claims to dividends, which are paid in units of the consumption good. The vector of dividends DD.D1; : : : ; Dds/ evolves according to
dDt DItDŒ .t; Dt; Yt/dtC .t; Dt; Yt/dWt (3.1) dYt DY.t; Yt/dtCY.t; Yt/dWt; (3.2) where ItD is a diagonal matrix with vector of dividends Dt on its diagonal, .t; Dt; Yt/ is the ds-dimensional vector of expected dividend growth rates and .t; Dt; Yt/ is the dsd volatility matrix of dividend growth rates. Likewise, Y.t; Yt/is thek-dimensional vector of expected changes in the state variables and Y .t; Yt/theirkd matrix of volatility coefficients. Coefficients of the stochastic differential equations (3.1)–(3.2) are assumed to satisfy standard conditions for the existence of a unique strong solution.D; Y /. Stocks are in unit supply (the number of shares is normalized to one). The aggregate dividend (aggregate consumption) is C10D.
Stock prices are determined in a competitive equilibrium. Equilibrium prices are assumed to have a representation
dStCDtdtDItSΠ.t; Dt; Yt/dtC .t; Dt; Yt/dWt ; (3.3) where .t; Dt; Yt/is theds-dimensional expected return and .t; Dt; Yt/theds d matrix of return volatilities. The coefficients .t; Dt; Yt/ and .t; Dt; Yt/are endogenous.
The locally riskless asset is a money market account which pays interest at some rater .t; Dt; Yt/per unit time. There is no exogenous supply of this asset (the money market account is an inside asset in zero net supply). The interest rate, representing the return on the asset, is also endogenously determined in equilibrium.
To simplify notation the arguments of drift, volatility and other functions are sometimes omitted. For examplertwill be used to denoter .t; Dt; Yt/,tto denote .t; Dt; Yt/, etc.
The following assumptions are made
Assumption 1. Candidate equilibrium prices processes satisfy the following conditions
(i) RT
0 jrvjdv<1; Pa:s:
(ii) RT
0 P
ijivj CP
i;jˇˇˇ vv0
i;jˇˇˇ
dv<1; Pa:s:
Assumption1is a set of restrictions on the space of candidate equilibrium prices processes. These assumptions are weak. Condition (i) ensures that the discount factor at the riskfree ratebt exp
Rt
0rvdv
is strictly positive for allt 2Œ0; T .
Condition (ii) ensures that the cumulative expected returns and return variances exist. This condition is sufficient for the existence of the total return process in (3.3).
3.2.2 Consumption, Portfolios and Wealth
The economy has a representative agent endowed with1share of each stock. The endowment of the money market account is null. The standing agent consumes and allocates wealth among the different assets available. Let Xt be the wealth at date t. Consumption is ct and t is the d1 vector of wealth proportions invested in the risky assets (thus1t01 is the proportion invested in the riskless asset). Consumption satisfies the physical nonnegativity constraintc 0. No sign restrictions are placed on the proportions invested in the various assets: long as well as short positions are permitted. The evolution of wealth is governed by the stochastic differential equation
dXtD.Xtrtct/dtCXtt0Œ.trt1/dtCtdWt (3.4) subject to some initial conditionX0 Dx 10S. For this evolutionary equation to make sense the following integrability condition is imposed on the policy.c; /
Z T
0
jctj CˇˇXtt0.trt1/ˇˇCˇˇXtt0tt0tXtˇˇdt<1; .Pa:s:/: (3.5) Under (3.5) the stochastic integral on the right hand side of (3.4) is well defined.
Condition (3.5) is a joint restriction on consumption-portfolio policies and candidate equilibrium price processes.
3.2.3 Preferences
Preferences are assumed to have the time-separable von Neumann-Morgenstern representation. The felicity provided by a consumption plan.c/is
U.c/E Z T
0 u.cv;v/dv ; (3.6)
where the utility function u W ŒA;1/Œ0; T ! Ris strictly increasing, strictly concave and differentiable over its domain. The consumption lower bound is assumed to be nonnegative,A 0. The limiting conditions limc!Au0.c; t/ D 1 and limc!1u0.c; t/D0are also assumed to hold, for allt 2Œ0; T . IfŒA;1/is a proper subset ofRC(i.e.A > 0) the function u is extended toRCŒ0; T by setting u.c; t/D 1forc2RCnŒA;1/and for allt 2Œ0; T .
This class of utility functions includes the HARA specification u.c; t/D 1
1R.cA/1R;
whereR > 0andA 0. IfA > 0the function has the required properties over the subsetŒA;1/RC. The function is then extended by setting u.c; t/ D 1 for0 c < A. This particular HARA specification corresponds to a model with subsistence consumptionA.
Under these assumptions the inverseIWRC Œ0; T !ŒA;1/of the marginal utility function u0.ct; t/ with respect to its first argument exists and is unique.
It is also strictly decreasing with limiting values limy!0I.y; t/D 1 and limy!1I.y; t/DA.
3.2.4 The Consumption-Portfolio Choice Problem
The consumer-investor seeks to maximize expected utility max.c;/U.c/E
Z T
0 u.cv;v/ dv (3.7)
subject to the constraints
dXt D.rtXtct/dtCXtt0Œ.trt/dtCtdWtI X0Dxi (3.8)
ct 0; Xt 0 (3.9)
for allt 2 Œ0; T , and the integrability condition (3.5). The first constraint, (25.6), describes the evolution of wealth given a consumption-portfolio policy.c; /. The quantityxrepresents initial resources, given by the value of endowmentsxD10S0. The next one (25.7) has two parts. The first captures the physical restriction that consumption cannot be negative. The second is a non-default condition requiring that wealth can never become negative.
A policy.c; / is said to be admissible, written.c; / 2 A, if and only if it satisfies (25.6) and (25.7). A policy.c; /is optimal, written.c; /2 A, if and only if it cannot be dominated, i.e.,U.c/U.c/for all.c; /2A.
3.2.5 Equilibrium
A competitive equilibrium is a collection of stochastic processes.c; ; S0; r; ; / such that:
1. Individual rationality:.c; /2A, where.S0; r; ; /is taken as given.
2. Market clearing: (a) commodity market: cDC10D, (b) equity market:
XDSand (c) money market:01D1.
This notion of equilibrium involves rational expectations. The representative agent correctly forecasts the evolution of asset returns when making individual decisions (condition 1). In equilibrium, forecasted return processes and market clearing return processes coincide (condition 2).
For later developments it is also useful to note that clearing of the commodity market implies clearing of the equity and money markets.