New utility functions

Given problems with the consumption-based model, the most natural place to start is by questioning the utility function. Functional form is not really an issue, since linearized and nonlinear models already behave similarly. Different arguments of the utility function are a more likely source of progress. Perhaps the marginal utility of consumption today depends on variables other than today’s consumption.

To get this effect, the utility function must be non-separable. If a utility function is separable, u(c, x) = v(c) + w(x), then ∂u(c, x)/∂c = v⁰(c) and x does not matter. This is the implicit assumption that allowed us to use only nondurable consumption rather than total consumption in the first place. To have marginal utility of consumption depend on something else, we must have a functional form that does not add up in this way, so that

∂u(c, x)/∂c is a function of x, too.

The first place to look for nonseparability is across goods. Perhaps the marginal utility of nondurable consumption is affected by durables, or by leisure. Also, business cycles are much clearer in durables purchases and employment data, so business-cycle risk in stock returns may correlate better with these variables than with nondurable and services consumption.

One problem with this generalization is that we don’t have much intuition for which way the effect should go. If you work harder, does that make a TV more valuable as a break from all that work, or less valuable since you have less time to enjoy it? Thus, will you believe an estimate that relies strongly on one or the other effect?

We can also consider nonseparability over time. This was always clear for durable goods.

If you bought a car last year, it still provides utility today. One way to model this nonsep-arabitlity is to posit a separable utility over the services, and a durable goods stock that depreciates over time;

U =X

t

β^tu(kt); kt+1= (1− δ)k^t+ ct+1.

This expression is equivalent to writing down a utility function in which last year’s purchases give utility directly today,

If u (·) is concave, this function is nonseparable, so marginal utility at t is affected by con-sumption (purchases) at t− j. At some horizon, all goods are durable. Yesterday’s pizza lowers the marginal utility for another pizza today.

Following this line also leads us to thinking about the opposite direction: habits. If good times lead people to acquire a “taste for the good life,” higher consumption in the past might raise rather than lower the marginal utility of consumption today. A simple formulation is to introduce the “habit level” or “subsistence level” of consumption xt, and then let

U =X

Again, you see how this natural idea leads to a nonseparable utility function in which past consumption can affect marginal utility today.

A difficulty in adding multiple goods is that, if the nonseparability is strong enough to affect asset prices, it tends to affect other prices as well. People start to care a lot about the composition of their consumption stream. Therefore, if we hold quantities fixed (as in the endowment-economy GMM tradition), such models tend to predict lots of relative price and interest-rate variation; if we hold prices fixed such models tend to predict lots of quantity variation, including serial correlation in consumption growth. An investigation with multiple goods needs to include the first order condition for allocation across goods, and this often causes trouble.

Finally, utility could be nonseparable across states of nature. Epstein and Zin (1991) pioneered this idea in the asset-pricing literature. The expected utility function adds over

states, just as separable utility adds over goods, Eu(c) =X

s

π(s)u [c(s)]

Epstein and Zin propose a recursive formulation of utility Ut=

(I use Hansen, Heaton and Li’s (2005) notation) that among other things abandons separabil-ity across states of nature. The term£

Et

¡U_t+1^1−γ¢¤_1−γ¹

is sometimes called a “risk adjustment”

or the “certain equivalent” of future utility. The Epstein-Zin formulation separates the co-efficient of risk aversion γ from the inverse of the elasticity of intertemporal substitution ρ.

Equation (13) reduces to power utility for ρ = γ. Models with non-time separable utilities (habits, durables) also distinguish risk aversion and intertemporal substitution, but not in such a simple way.

The stochastic discount factor/marginal rate of substitution is

mt+1 = β

(The appendix contains a short derivation.) If ρ6= γ, we see a second term; expected returns will depend on covariances with changes in the utility index, capturing news about the investor’s future prospects, as well as on covariances with consumption growth. As we will see, a large number of modifications to the standard setup lead to a marginal rate of substitution that is the old power formula times a multiplicative new piece.

The utility index itself is not directly measurable, so to make this formula operational we need some procedure for measurement. It turns out that the utility index is proportional to the value of the wealth portfolio (the claim to the consumption stream), so one can write the discount factor

(This formula is also derived in the appendix.) This effect provides a route to including stock returns in the asset pricing model alongside consumption growth, which of course can give a much improved fit. This was the central theoretical and empirical point of Epstein and Zin (1991). However, this modification stands a bit on shaky ground: the substitution only works for the entire wealth portfolio (claim to future consumption), including nontraded assets such as real estate and the present value of labor income, not the stock market return alone.

Furthermore, wealth and consumption do not move independently; news about consumption growth moves the wealth return.

To emphasize the latter point, Restoy and Weil (1988, p. 10) show how to approximate

the wealth return by a discounted sum of future consumption, leading to a formula valid near ρ = 1,

(Et+1− E^t) log mt+1 ≈ (E^t+1− E^t)

"

−ρ∆ log c^t+1+ (ρ− γ) X∞

j=0

β^j∆ log ct+j

#

(16)

where small letters denote logs of capital letters. Future long-horizon consumption growth enters the current period marginal rate of substitution. In essence, this formulation tracks the wealth-portfolio return or utility index term back to its source in future values of con-sumption. Thus, variables that predict future consumption growth will appear as additional risk factors even with (perfectly measured) current consumption growth.