Л. Intertemporal Equilibrium Models

depend on die size ol a Hade. When dicre are transactions costs, the after-iransaction-cosi return on an asset bought today and sold tomorrow is not the negative of the aliei-li ans.u lion-cost relurn on the same asset sold today and bought back tomorrow. These two returns can be measured separately and can both be included in the sol of returns if they are made subject to shot (sales const taints.

In the presence of shorlsales constraints, the vector equality (H. 1.8) is replaced by another vector equality

I) = K</.+ R,)A7,], (8.3.2)

where 0 is an unknown vector. The model implies various restrictions on 0 such as the restriction lhat 0, < 1 for all i. Volatility bounds can now be lotiiid for each Л! by choosing, subject to the restrictions, the value of 0 that delivers the lowest variance for Л7*(Л?). He and Modest (1095) find that by combining hot rowing constraints, a restriction on the short sale of Treasury bills, and asset-specific transaction costs they can greatly reduce llicvolatility hound on the stochastic discount factor.

This analysis is extremely conservative in that в is chosen to minimize the volatility hound without asking what underlying equilibrium would support this choice for 0. If there are substantial transactions costs, for example, then even risk-neutral traders will not sell one asset lo buy another asset with a higher relurn unless the return difference exceeds the transactions costs. Hut the one-period transaction costs will not be relevant if traders can buy the high-return asset and hold it for many periods, or if a trader has new wealth to invest and must pay the cost of purchasing one asset or the other. Tims the work of He and Modest (1995) and l.nttmer (1994) is exploratory, a way to gel a sense for the extent lo which market frictions loosen the bounds implied by a Ii ictionless market.

Some authors have tried lo solve explicitly for the asset prices that arc implied by equilibrium models with transactions costs. This is a difficult task because transactions costs make the investors decision problem comparatively intractable except in very special cases (see Davis and Norman (1990)). Aivagari and (order (1991), Amihiid and Mendelson (1980), Conslanthiides (1980), I leaton and l.ucas (1990), and Vayanos (1995) have begun to make some progress on ibis topic.

V. 1.2 Maihcl Frictions anil Aggregate Consumption Data

The rejection of the standard consumption CAPM may be due in part lo difficulties in measuring aggregate consumption. The consumption ( АРМ applies to true consumption measured at a point in time, but the available data are lime-aggrogalcd and measured with error. Wilcox (1992) describes the sampling procedures used to const tint consumption data, while Grossman,

8.3. Market Frictions

:, <

Melino, and Shiller (1987) and Wheatley (1988) have tested the model] lowing for time-aggregation and measurement error, respectively. Roughly] speaking, these data problems can cause asset returns weighted by measured marginal utility of consumption, (\+R,J+\)S {C,+i / C,)~r, to be forecastaple . in the short run but not the long run. Thus one can allow for such problems by lagging the instruments more than one period when testing the model.12 Doing this naturally weakens the evidence against the consumption CAPM, but the model is still rejected at conventional significance levels unless very long lags are used. j

A more radical suggestion is that aggregate consumption is not an adequate proxy for the consumption of stock market investors even in the long run. One simple explanation is that there are two types of agents in the economy: constrained agents who are prevented from trading in asset markets and simply consume their labor income each period, and unconstrained agents. The consumption of the constrained agents is irrelevantjto the determinatioirof equilibrium asset prices, but it may be a large fraction of aggregate consumption. Campbell and Mankiw (1990) argue that predictable variation in consumption growth, correlated with predictable variation in income growth, suggests an important role for constrained agents, while Mankiw and Zeldes (1991) use panel data to show lhat the consumption of-stockholders is more volatile and more highly correlated with the stock market than the consumption of nonstockholders.

The constrained agents in the above model do not directly influence asset prices, because they are assumed not to hold or trade financial assets. Another strand of the literature argues that there maybe some investors who buy and sell stocks for exogenous, perhaps psychological reasons. These noise traders can influence stock prices because other investors, who are rational utility-maximizers, must be induced to accommodate their shifts in demand. If utility-maximizing investors are risk-averse, then they will only buy stocks from noise traders who wish to sell if stock prices fall and expected stock returns rise; conversely they will only sell stocks to noise traders who wish to buy if stock prices rise and expected stock returns fall. Campbell and Kyle (1993), Cutler, Poterba, and Summers (1991), DeLong etal. (1990a, 1990b), and Shiller (1984) develop this model in some detail. The model implies that rational investors do not hold the market portfolio- instead they shift in and out of the stock market in response to changing demand from noise traders-and do not consume aggregate consumption since some consumption is accounted for by noise traders. This makes the

Campbell and Mankiw (19941) discuss diis in the context of a linearized model. Breeden, Gibbons, anil I-itzenberger (1989) make a related point, arguing that at short horuons one should replace consumption with ihe relurn on a portfolio constructed to be highly correlated with longer-run movements in consumption. Brainard, Nelson, and Shapiro (1991) find lhat the consumption ( .АРМ works better at long horizons than at short horizons.

£ Ж

model hard to test without having detailed information on the investment strategies of different market participants.

It is also possible lhat utility-maximizing stock market investors are heterogeneous in important ways. If investors arc subject to large idiosyncratic risks in their labor income and can share these risks only indirectly by trading Ya few assets such as stocks and Treasury bills, their individual consumption paths may be much more volatile than aggregate consumption. Even if individual investors have the same power utility function, so that any individuals consumption growth rate raised to the power - у would be a valid stochastic discount factor, the aggregate consumption growth rate raised to the power

у may not be a valid stochastic discount factor.1 Problem 8.3, based on Mankiw (1986), explores this effect in a simple two-period model.

Recent research has begun to explore the empirical relevance of imperfect risk-sharing for asset pricing. Healon and Lucas (1996) calibrate individual income processes to micro data from the Panel Study of Income bynam.ics (PSID). Because the PSID data show lhat idiosyncratic income variation is largely transitory, Healon and I.ucas find that investors can min-mize its effects on their consumption by borrowing and lending. Thus they ind only limited effects on asset pricing unless they restrict borrowing or issume the presence of large transactions costs. Constantinidcs and Duffie

(1996) construct a theoretical model in which idiosyncratic shocks have per-nancnt effects on income; they show that this type of income risk can have large effects on asset pricing.

! Given this evidence, it seems important lo develop empirically testable inlerlemporal asset pricing models thai do not rely so heavily on aggregate consumption data. One approach is to subslitute consumption out of the consumption CAPM lo obtain an asset pricing model that relates mean returns to covariances with the underlying state variables that determine consumption. The strategy is to try to characterize the preferences that an investor would have to have in order to be willing to buy and hold the aggregate wealth portfolio, without necessarily assuming thai this investor also consumes aggregate consumption.

There arc several classic asscl pricing models of ibis type set in continuous time, most notably Cox, Ingcrsoll, and Ross (1985a) and Merton (1973a). But those models are hard to work with empirically. Campbell (1993a) suggests a simple way to get an empirically tractable discrete-time

This is an example ol Jensens Inequality. Since marginal utility is nonlinear, the average ot investors marginal utilities ot consumption is not generally the same as the marginal utility ol average consumption. This problem disappears when investors individual consumption streams are perfectly correlated with one another as they will be in a complete markets setting. Grossman and Shiller (19H2) point out that it also disappears in a continuous-lime model when the processes for individual consumption streams and asset prices are diffusions.

1 = E,

C-m-i c,

(1 + /< ,.,+ ,)

(1 + /i,./+i)

(8.3.5)

If we assume that asset returns and consumption are homoskedastic and jointly lognormal, ihen this implies that the riskless real interest rale is

r + 1 = -l RS + al - --a; + yV.,li\fH,l (8.3.6)

J. 2у>- у/

There are in fail typos in equations (1(1) lluongb (12) of Kpslcin anil /.in (1991) which give intermediate steps in the (lei ivaliotl.

model using the utility specification developed by Epstein and /m (1989, 1991) and Weil (1989), which we now summarize.

Separating lli.sk Aversion and Inlerlemporal Substitution

Epstein, /.in, and Weil build on the approach of Keeps and Poi tens (1978) lo develop a more flexible version of the basic power utility model, lhat model is restrictive in that it makes the elasticity of intertemporal substitution, i/, the reciprocal of the coefficient ol relative risk aversion, у. Ye I it is not clear that these two concepts should be linked so tightly. Risk aversion describes the consumers reluctance to substitute consumption across slates of die world and is meaningful even in an alempoial setting, whereas the elasticity of inlet temporal substitution describes the consumers willingness lo sul>-slitute consumption over time and is meaningful even in a deleiininistic setting. Ilie Epsteiu-/in-Weil model retains many ol the attractive features of power milily but breaks the link between ihe parameters у and ф. Ihe Epsteiu-/in-Weil objective function is defined recursively by

V, = J (1 - 8)(y + S (E, [Utf l) J , (8.3.3)

where fV = (I - y)/(\ - 1/y/). When I) = I the recursion (8.3.3) becomes linear; it can then be solved forward lo yield the familiar time-separable power utility model.

The intertemporal budget constraint for a representative agent can be written as

Wl+, = (I + I{mJ+i)(W, - C), (8.3.4)

where W,+1 is (he representative agents wealth, and (I + />, /+l) is the return on the market portfolio of all invested wealth. This form of the budget constraint is appropriate for a complete-markets model in which wealth includes human capital as well as financial assets. Epstein and /in use a complicated dynamic programming argument lo show that (8.3.3) ami (8.3.4) together imply an Ettlcr equation of the form11

The premium on riskv assets, including the market portfolio itself, is

K,rlMll - v.,h + - = в ~ + (1 -t>)rr,w. (8.3.7)

lliis says lliat tlie risk premium on asset i is a weighted combination of asset is covariance with consumption growth (divided by the elasticity of intertemporal substitution ф) and asset is covariance with the market return. The weights are 0 and I - 0 respectively. The Epslein-Zin-Weil model thus nests the consumption CAPM with power utility (0 = 1) and the traditional static CAPM (t> = 0).

It is templing to use (8.3.7) together with observed data on aggregate consumption and stock market returns to estimate and test the Epstcin-Zin-Weil model. Epstein and Zin (1991) report results of this lype. In a similar spirit, Giovanniui and Weil (1989) use ihe model to reinterpret the results of Mankiw and Shapiro (198b), who found that betas will, the market have greater explanatory power for the cross-sectional pattern of returns than do betas with consumption; this is consistent with a value old close lo /его. However (his procedure ignores the fact lhat the intertemporal budget constraint (8.3.1) also links consumption and market returns. We now show that the budget constraint can be used lo substitute consumption out of the asset pricing model.

Substituting Consumption Out of the Model

Campbell (1993a) points out that one can loglineari/e the intertemporal budget constraint (8.3.1) around the mean log consumption-wealth ratio to obtain

A ic i ~ i ,mi + ><+ - (0 - ,). (8.3.8)

where /) = 1 - oxp(< - w) and It is a constant that plays no role in what follows. (amihiiiing ibis will) die trivial equality Лк/ц = A<i+1 ~ A(,+ l - и ,.,.i), solving the resulting difference equation forward, and taking expectations, we can write die budget constraint in the form

ui.i+i

A0+;1

(8.3.9)

This equation savs that if the consumption-wealth ralio is high, then the agent must expect either high returns on wealth in the future or low consumption growth rails. This follows just from the approximate budget constraint without imposing ai iv behavioral assumptions. Il is directly analogous

to the linearized formula for the log dividend-price ratio in Chapter 7. Here wealth can be thought of as an asset that pays consumption as its dividend.15 If we now combine the budget constraint (8.3.9) with the loglinearEuler equations for the Epstein-Zin-Weil model, (8.3.6) and (8.3.7), we obtain a closed-form solution for consumption relative to wealth:

r, - w, = (1 - f)K,

L.PJr<n.+j

P{k-Hm) \-p

(8.3.10)

Here /*, is a constant related to the conditional variances of consumption growth and the market portfolio return. The log consumption-wealth ratio is a constant, plus (1 - rp) times the discounted value of expected future returns on invested wealth. If ф is less than one, the consumer is reluctant to substitute intertemporally and the income effect of higher returns dominates the substitution effect, raising todays consumption relative to wealth. If ф is greater than one, the substitution effect dominates and the consumption-wealth ratio falls when expected returns rise. Thus (8.3.10) extends to a dynamic context the classic comparative statics results of Samuelson (1969).

(8.3.10) implies that the innovation in consumption is

ci+t-Ei[fi+i] = rm +i - E,[rml+\] + (l-Vfr)(E,+I

(8.3.11)

L;=i

An unexpected return on invested wealth has a one-for-one effect on consumption, no matter what the parameters of the utility function: This follows from the scale independence of the objective funcdon (8.3.3). An increase in expected future returns raises or lowers consumption depending on whether ф is greater or less than one. Equation (8.3.11) also shows when consumption will be smoother than the return on the market. When the market return is mean-reverting, there is a negative correlation between current returns and revisions in expectations of future returns. This reduces the variability of consumption growth if the elasticity of intertemporal substitution ф is less than one but amplifies it if ф is greater than one.

Equation (8.3.11) implies that the covariance of any asset return with consumption growth can be rewritten in terms of covariances with the re-

Campbell (1993a) anil Campbell anil Koo (199b) explore the accuracy of the loglinear approximation in this context by comparing the approximate analytical solution for optimal consumption with a numerical solution. In an example calibrated to US stock market data, the two solutions are close together provided that the investors elasticity of intertemporal substitution is less than about 3.