8. Intertemporal Equilibrium Models

In the internal-habit sK-t ideation, die derivation of die Kuler equation is omplii atcd by ill** fail dial linit--/ consumption affects die summation in (8. I.I) through die term dated / I I as well as the lenn dated /. We have

iurjdc, = [i o\(cni/c,)1 >(Xn]/xl),-t](cl/x,)]->(\/с,), [нал)

This is random al lime / because il depends on consumplion al lime i-H. Substituting in for X, and imposing die condition that the agents own consumption equals aggregate consumption, this becomes

If this model is to capture the idea of habit formation, then we need к(у - 1) > 0 ю ensure thai an increase in yesterdays consumption increases the marginal utility of consumplion today. The Kuler equation can now be written as

к, I ;>( .к:,] = sv.,\(\ + nl+iydUl+[/iH:,+l\, (8.4.9)

where the expectations operator on the left-hand side is uecessaiy because ol the randomiless of <) / )(.(.

The analysis simplifies considerably in the external-habit specification. In ibis case (8.4.8) and (K.4/.I) can be combined lo give

I = M,(l-f/Ui)<W-i)MrAWQ~]. (8.4.10)

If we assume homoskedasticily and joint lognoi nudity of asset returns and consumption growlh, (his implies the following restrictions on risk premia and the riskless real interest rale:

/./it = -log1* - кУ7- -f кК,Ас,+ 1] - к(у - 1)Дг (8.4.11)

l/Ii./i-i - /.nil = yo . (8.4.12)

Equation (8.4.11) says that the riskless real interest rale equals its value under power utility, less к[у - I )Д<,. 1 lolding consumption today and ex-peeled consumption tomorrow constant, an increase in consumption ves-tctday increases the marginal uiility of consumption today. This makes the representative agent want to borrow from the future, driving up the real interest rale. Kquation (8.4.12) describing ihe risk premium is exactly the same as (8.2.7), the risk premium formula for the power utility model. The external habit simply adds a lenn lo the Kuler equation (8.4.10) which is known al lime /, and this does not affect the risk premium.

Abel (W.M. l.)O(i) nevertheless argues thai catching up wilb the Joneses can help lo explain the equiiy premium puzzle. This argument is based on Iwo considerations, first, the average level of the riskless rale in (8.4.11) is

8.4. More General Utility Functions

- log S-y2af/2+(y-к(у -1 ))g, where g is the average consumption growth rate. When risk aversion у is very large, a positive к reduces the average riskless rate. Thus catching up with the Joneses enables one to increase risk aversion lo solve the equity premium puzzle without encountering the riskfree rate puzzle. Second, a positive к is likely to make the riskless real interest rate more variable because of the term - к(у - 1)Дс, in (8.4.11). If one solves Tor the stock returns implied by the assumption that stock dividends equal consumplion, a more variable real interest rate increases the covariance of stock returns and consumption aic and drives up the equity premium.

The second of these points can be regarded as a weakness rather than a strength of the model. The equity premium puzzle shown in Table 8,1 is that the ratio of the measured equity premium to the measured covariajnee crlc is large; increasing the value a-u implied by a model that equates stpek dividends with consumption does not improve matters. Also the real interest rate does not vary greatly in the short run; the standard deviation of the ex post real commercial paper return in Table 8.1 is 5.5%, and Table 8.2 shows thai about a third of the variance of this return is forecastable, implying a standard deviation for the expected real interest rate of only 3%. Since the standard deviation of consumption growth is also about 3%, large values of к and у in equation (8.4.11) tend to produce counterfactual volatility! in the expected real interest rate. Similar problems arise in the internal-hajbit model. j

This difficulty with the riskless real interest rate is a fundamental problem for habit-formation models. Time-nonseparable preferences make marginal utility volatile even when consumption is smooth, because consumers derive utility from consumption relative to its recent history rather than from the absolute level of consumption. But unless the consumption and habit processes take particular forms, time-nonseparability also creates large swings in expected marginal utility at successive dates, and this implfes large movements in the real interest rate. We now present an alternative specification in which it is possible to solve this problem.

Difference Models

Consider a model in which the utility function is

U, = E,

ri ici+)- Xi+j) Y ~ 1

(8.4.13)

and for simplicity treat the habit level X, as external. This model differs from the ratio model in two important ways. First, in the difference model the agents risk aversion varies with the level of consumption relative to habit, whereas risk aversion is constant in the ratio model. Second, in the

difference model consumption must always be above habit for utility to be well-defined, whereas this is not required in the ratio model.

To understand the first point, it is convenient lo work with the surplus consumption ratio S defined by

S, s (8.4.14)

The surplus consumption ratio gives the fraction of total consumption that is surplus to subsistence or habit requirements. If habit X, is held fixed as consumption C, varies, the normalized curvature of the utility function, which would equal the coefficient of relative risk aversion and would be a constant у in the conventional power utility model, is

-CUcf: Y (8.4.15)

uc. St

This measure of risk aversion rises as the the surplus consumption ratio St declines, that is, as consumption declines toward habit.17

The requirement that consumption always be above habit is satisfied automatically in microcconomic models with exogenous asset returns and endogenous consumption, as in Constantinides (1990) and Sundarcsan (1989). It presents a more serious problem in models with exogenous consumption processes. To handle this problem Campbell and Cochrane (1995) specify a nonlinear process by which habit adjusts to consumption, remaining below consumption at all times. Campbell and Cochrane write down a process for the log surplus consumption ratio s, m log(.S,). They /fssumc that log consumption follows a random walk with drift g and innovation Дс,+1 = g + They propose an AR(1) model for s,:

j s,+i = (1 -ф)1 + Ф-п + Х(5,)ип1. (8.4.16)

Here 1 is the steady-state surplus consumption ratio. The parameter ф governs the persistence of the log surplus consumption ralio, while the sensi-ivity function X(s,) controls the sensitivity of st+\ and thus of log habit x,+i i о innovations in consumption growth

Equation (8.4.16) specifics that todays habit is a complex nonlinear function of current and past consumption. By taking a linear approximation around the steady state, however, it may be shown-that (8.4.16) is ap-

l7Risk aversion may also be measured by the normalized curvature of the value function i maximized utility expressed as a function of wealth), or by the volatility of the stochastic iscomu factor, or by the maximum Sharpe ratio available in asset markets. While these ijieasurcx of risk aversion are different from each other in this model, they all move inversely villi Л,. Note that y, the curvature parameter in utility, is no longer a measure of risk aversion l this model.

proximately a traditional habit-lot tnntion model in which log habit responds slowly to log c:onsnmplion,

л/+) % [(1 - ф) h -t- у-] + фх, -Mi - ф) <;

+ (1 -</)]Г Фс,-,. (8.4.17)

where h = lu( 1 - S) is the steady stale value of x-r. The problem with die traditional model (8.4.17) is that it allows consumption to fall below habit, resulting in infinite or negative marginal utility. Л process for st defined over the real line implies lhat consumption can never fall below habit.

Since habit is external, the marginal utility of consumption is u(C,) = (С, - X,)-y = ,V, r С, y. The stochastic discount lactor is then

.. .m(C,+i) / Sl+1 Cl+1 \ Y

= S rs = 5Hr 7~ (8.4.18)

и(С,) \ S, С, J

In the standard power utility model .V, = 1, so the stochastic discount factor isjust consumption growth raised to the power-y. To get a volatile stochastic discount factor one needs a large value of у. In the habit-formation model one can instead get a volatile stochastic discount factor from a volatile surplus consumption ratio St.

The riskless real interest rale is related to the stochastic discount factor by (1 + = \/E,{Ml+i). Taking logs, and using (8.4.10) and (8.4.18),

the log riskless real interest rale is

r/+1 = - log(S) + yg~ y(\ - ф){я, - л) - p- (Л(л,) 4- If (8.4.19)

The first two lerms on the right-hand side of (8.4.19) are familiar from the power utility model (8.2.6), while die last Iwo terms are new. The third term (linear in (s, - 1)) reflects intertemporal substitution, or mean-reversion in marginal utility. If the surplus consumption ratio is low, the marginal utility of consumption is high. However, the surplus consumption ratio is expected to revert to its mean, so marginal utility is expected to fall in the future. Therefore, the consumer would like to borrow and this drives up the equilibrium risk free interest rate. The fourth term (linear in iA.(.f,) + l]2) reflects precautionary savings. As uncertainty increases, consumers become more willing to save and this drives down the equilibrium riskless interest rate.

If this mode! is lo generate stable real interest rates like those observed in the data, the serial correlation parameter ф must be near one. Also, die sensitivity function Л(л,) must decline with s, so that uncertainly is high when

1-0.

. lujHiKu i.quiiuniuin iMottets

v, is low and (he precautionary saving lenn olfsets ihe inlerlemporal sttbstitu-lion lenn. In lad, Cam>hell and Cochrane parametrize the A(.v,) (unction so thai these iwo terms exadly offset each other everywhere, implying a constant riskless interest rale.

Even with a constant riskless interest rate and random-walk consumption, the external-habit model can produce a large equity premium, volatile-stock prices, and predictable excess slock returns. The basic mechanism is time-vat iatiou in risk aversion. When consumption falls relative lo habit, the resulting increase in risk aversion drives up the risk premium on risky assets such as slocks. This also drives down the prices of stocks, helping to explain why slock returns are so much more volatile than consumplion growth or riskless real interest rales.

Campbell and Cochrane (1995) calibrate their model to USdat a on consumption and dividends, solving for equilibrium slock prices in the tradition of Mehra and Prescoii (1985). There is also some work on habit formation that uses actual slock return dala in the tradition of Hansen and Singleton (1982, IHS:*). Healon (1995), for example, estimates an iniernal-habil model allowing for lime-aggregation of the data and for some durability of those goods formally described as nondurable in the national income accounts. Durability can he thought of as the opposite of habit formation, in that consumption expenditure today lowers the marginal utility of consumplion expenditure tomorrow. Healon finds that durability predominates at high frequencies, and habit formation al lower frequencies. However his habit-formation model, like the simple power utility model, is rejected statistically.

Both these approaches assume dial aggregate consumplion is the driving process for marginal utility. An alternative view is thai, for reasons discussed in Section K.:V>, the consumplion of slock market investors may not be adequately proxied by macroeconomic data on aggregate consumplion. Under this view the driving process lor a habit-formation model should be a process with a reasonable mean and standard deviation, but need not be highly correlated with aggregate consumplion.

Л. 7.2 Psychological Models of Preference*

lsychologisls and experimental economists have found that in experimental settings, people make choices that differ in several respects from the standard model ol expected utility. In response lo these findings unorthodox psychological models of preferences have been suggested, and some recent research has begun lo apply lliese models lo asset pricing. 1

Usi-lnl цси.-ial iclctcnicsimliiilc 11..Ц.П ih anil Rcder (НЖ7) anil Keeps (I.IHH).

HA. More Central Utility Functions

Psychological models may best be understood by comparing them to th standard time-separable specification (8.1.1) in which an investor maximizes

X>£AC(+y)

(8.4.20)

This specification has three main components: the period utility function (7(C), the geometric discounting with discount factor 5, and the mathematical expectations operator E,. Psychological models alter one or more of these components.

The best-known psychological model of decision-making is probably the prospect theory of Kahneman andTversky (1979) and Tversky and Kahneman (1992). Prospect theory was originally formulated in a static context, so it does hot emphasize discounting, but it does alter the other two elements of the standard framework. Instead of defining preferences over consumption, preferences are denned over gains and losses relative to some benchmark outcome. A key feature of the theory is that losses are given greater weight than gains. .Thus if x is a random variable that is positive for gains and negative for losses, utility might depend on

t-i-i

ifx > 0

ti(x)- = , (8.4.21)

I l-W !

Here у\ an-d уг are curvature parameters for gains and losses, which may differ from one another, and A. > 1 measures the extent of loss aversion, the greater weight given to losses than gains.

Prospect theory also changes the mathematical expectations operator in (8.4.20). The expectations operator weights each possible outcome by its probability; prospect theory allows outcomes to be weighted by nonlineaij functions of their probabilities (see Kahneman and Tversky (1979)) or b)j nonlinear functions of the probabilities of a better or worse outcome. Other; more general models of investor psychology also replace the mathematica expectations operator with a model of subjective expectations. See for е* ample Barberis, Shleifer, and Vishny (1996) DeLong, Shleifer, Summers and Waldmann (1990b), and Froot (1989).

In applying prospect theory to asset pricing, a key question is how the benchmark outcome defining gains and losses evolves over time. Benartzii and Thaler (1995) assume that investors have preferences defined overj returns, where a zero return marks the boundary between a gain and al loss. Returns may be measured over different horizons; a /(-month return! is relevant if investors update their benchmark outcomes every К months. Benartzi and Thaler consider values of К ranging from one to 18. They show that loss aversion combined with a short horizon can rationalize investors