Asset Prices under Habit Formation and Catching up with the Joneses

Introduce a utility function nesting three classes of preferences and use it, inside a Lucas asset pricing model, to derive closed-form asset prices and examine the equity premium puzzle (Abstract and p. 1, NBER Working Paper 3279). The three classes are “(1) time-separable utility functions; (2) ‘catching up with the Joneses’ utility functions that depend on the consumer’s level of consumption relative to the lagged cross-sectional average level of consumption; and (3) utility functions that display habit formation.” Incorporating this “into a Lucas (1978) asset pricing model allows calculation of closed-form solutions for the prices of stocks, bills and consols under the assumption that consumption growth is i.i.d. Then equilibrium asset prices are used to examine the equity premium puzzle.”

Through a preference parameter that is a geometric average of own and aggregate lagged consumption, weighted by D and raised to a power (Section I, pp. 1-2). If the power is zero the preference parameter is identically one and utility is time-separable. “If [the power is positive] and D = 0, the parameter [the benchmark] depends only on the lagged level of aggregate consumption per capita. This formulation is the relative consumption model or ‘catching up with the Joneses.’” And “if [the power is positive] and D = 1, the parameter [the benchmark] depends only on the consumer’s own past consumption. This formulation is the habit formation model.” Period utility is isoelastic in the ratio of consumption to the benchmark; “when [the power] = 0, the utility function … is the standard constant relative risk aversion utility function and [the curvature parameter] is the coefficient of relative risk aversion. More generally, utility depends on the level of consumption relative to some endogenous time-varying benchmark.”

Because the benchmark is the lagged, not the current, level of aggregate consumption (footnote 1, p. 12). “The phrase ‘catching up with the Joneses’, rather than ‘keeping up with the Joneses’, reflects the assumption that consumers care about the lagged value of aggregate consumption.” The same footnote records that an April 1989 version of Jordi Galí’s paper – though not its September 1989 revision – examines a related utility function and shows that for a particular parameter value asset pricing would be equivalent to an economy without consumption externalities and with log utility.

Only under habit formation does today’s consumption change the marginal utility of consumption tomorrow, adding a forward-looking term to the marginal utility of consumption (Section I, pp. 2-3, and Section II, p. 3). Differentiating lifetime utility with respect to current consumption holding aggregate consumption unchanged yields the usual within-period marginal utility plus a term proportional to D. In equilibrium this collapses into a factor which “equals 1 if [the product of the power and D] equals 0, which is the case for both time-separable and relative consumption preferences.” The paper reports a sufficient condition for that marginal utility to remain positive under habit formation: for a discount factor of 0.99 and the two-point distribution used in Table 1, “the sufficient condition is 0.858 < [the curvature parameter] < 1.142” (footnote 3, p. 12).

A Lucas exchange economy: output is a perishable consumption good produced by the capital stock, all output is consumed in the period it is produced, and all consumers are identical (Section II, p. 3). “Because all consumers are identical, [own consumption equals aggregate consumption equals output] in every period.” Gross output growth therefore equals the growth rate of both individual and aggregate consumption, and the growth of the preference benchmark is that same growth rate raised to the power governing the strength of the benchmark effect.

Stocks, one-period riskless bills and consols, all from the standard Euler condition that the conditional expectation of the product of the intertemporal marginal rate of substitution and the gross return equals one (Sections III-V, pp. 4-6). The stock is a claim to a unit of risky capital with an ex-dividend price, and the price-dividend ratio satisfies a forward-looking equation; the bill costs a price today and pays one unit of consumption next period; the consol pays one unit of consumption each period at an ex-coupon price. Abel notes a subtlety in the Euler condition itself: “In the conventional time-separable formulation of this problem, [the marginal utility of current consumption] is known as of time t, and hence [its conditional expectation] on the left hand side of (8) equals” that marginal utility (footnote 4, p. 12) – which is not generally true here, since the benchmark under habit formation makes current marginal utility depend on future variables.

Explicit solutions for all three prices, and closed-form unconditional expected returns for two of the three preference cases (Section VI, pp. 6-7). “Suppose that consumption growth … is i.i.d. over time. In this case, we can obtain explicit solutions for the prices of stock, bills, and consols.” The price-dividend ratio, bill price and consol price all take the form of a constant times a power of current growth divided by a common factor. “Given a distribution for x, the moments of x can be calculated and the three asset prices are easily calculated. For time-separable preferences … and relative consumption …, we can obtain closed-form solutions (in terms of preference parameters and the moments of x) for the unconditional expected returns,” while “under habit formation, unconditional expected returns can be calculated numerically using the asset prices.”

Mehra and Prescott’s finding that a calibrated time-separable isoelastic model could not deliver more than a 35 basis point equity premium against an observed 618 (Section VII, pp. 7-8). “Mehra and Prescott (1985) report that from 1889 to 1978 in the United States, the average annual real rate of return on short-term bills was 0.80% and the average annual real rate of return on stocks was 6.98%. Thus the average equity premium was 618 basis points.” Their model used “a 2-point Markov process for consumption growth with [mean] 1.018, [variance] (0.036)^2, and correlation … -0.14. For values of the preference parameters that Mehra and Prescott deemed reasonable, the model could not produce more than a 35 basis point equity premium … when the expected riskless rate … was less than or equal to 4% per year. This result is the equity premium puzzle.”

The same first two moments of consumption growth and a discount factor of 0.99, but i.i.d. growth rather than a serially correlated two-point Markov process (Section VII, p. 8, and Table 1, p. 11). “Table 1 reports the unconditional expected rates of return on stocks, bills, and consols under the assumption that [growth] is i.i.d, [with mean] 1.018 and [variance] (0.036)^2.” For the time-separable and relative consumption panels “two unconditional expected returns are reported in each cell: the first is calculated under a 2-point i.i.d. distribution; the second, which is in brackets, is calculated under a lognormal distribution for x.” The discount factor is 0.99. Note that Mehra and Prescott’s own process had a serial correlation of -0.14, which the i.i.d. assumption here sets aside.

The puzzle, reproduced: the equity premium never approaches the historical figure because the riskless rate rises with the stock return (Section VII, p. 8, and Table 1, p. 11). “The top panel of Table 1, which reports the unconditional expected rates of return under time-separable preferences, displays the equity premium puzzle. Although [the expected stock return] increases as [the curvature parameter] increases from 0.5 to 10.0, [the expected bill return] also increases. The equity premium … does not come anywhere close to the 600 point historical average.” The reported values run from 1.93 on stocks against 1.87 on bills at a curvature of 0.5, to 14.22 against 12.85 at a curvature of 10. Abel also notes an incidental property: “the unconditional expected rates of return of bills and consols are exactly equal under time-separable preferences.”

An equity premium of 463 basis points and a riskless rate of 2.07 percent at a curvature of 6 – much closer to the data – but with a conditional-return failure the paper reports immediately (Section VII, pp. 8-9, and Table 1, p. 11). “For [a curvature of] 6, the equity premium is 463 basis points and the unconditional riskless rate is 2.07% per year. Although the unconditional expected returns on stocks and bills are much closer to their historical averages, the conditional expected rates of return (not reported in the table) vary too much. For the 2-point distribution for x, the standard deviation of [the conditional expected bill return] is 17.87% when [the curvature parameter] = 6. This unrealistic implication of the model poses a challenge for future research.” The table also shows that raising the curvature to 10 pushes the expected stock return to 14.73 while the bill return falls to 1.59.

Extremely sensitive: expected returns on long-lived assets are very responsive to the curvature parameter, and exceed 35 percent at a curvature of 1.14 (Section VII, p. 9, and Table 1, p. 11). “The expected rates of return on both long-lived assets – stocks and consols – are extremely sensitive to the value of [the curvature parameter]. Under logarithmic utility …, the expected rates of return are the same as under time-separable preference and relative consumption. However, with [a curvature of] 1.14, the expected rates of return on stocks and consols are both greater than 35%.” The reported values move from 33.56 on stocks and 4.53 on bills at a curvature of 0.86, through 6.83 and 3.48 at 0.94 and the common logarithmic values of 2.83 and 2.70 at 1.00, to 8.43 and 1.93 at 1.06 and 38.28 and 0.93 at 1.14. The admissible range is narrow: the sufficient condition for positive marginal utility under this calibration is a curvature between 0.858 and 1.142, so the whole habit-formation panel sits inside a band of width less than 0.3.

Not substantially, at the parameter values reported (Section VII, p. 9). “The top and middle panels of Table 1 report unconditional rates of return for a lognormal distribution with [mean] 1.018 and [variance] (0.036)^2. For the parameter values reported, it makes no substantial difference for expected returns whether the growth rate is lognormal or has a 2-point distribution.” The bracketed figures in the table bear this out – for example, 10.33 against 10.34 on stocks at a curvature of 6 under time-separable preferences, and 6.72 against 6.70 at the same curvature under relative consumption.

Explore intermediate values of the two preference parameters and relax the i.i.d. assumption (Section VII, p. 9). “Further research using the utility function introduced in this paper will explore the implications of other settings for the parameters [governing the strength of the benchmark effect and its composition]. For instance, if D is between zero and one, the utility function would contain elements of both catching up with the Joneses as well as habit formation. Also the assumption of i.i.d. consumption growth rates can be relaxed, and asset prices can then be analyzed numerically.”

Alongside contemporaneous work on habit formation in asset pricing and on consumption externalities (footnote 2, p. 12, and References, p. 10). Abel notes that “George Constantinides (1988), Jerome Detemple (1989), John Heaton (1989), and Suresh Sundaresan (1989) also examine asset prices in the presence of habit formation. James Nason (1988) includes a time-varying benchmark level of consumption that differs from habit formation in that it is independent of an individual consumer’s own consumption” – that last distinction being precisely what separates the relative consumption case from habit formation in Abel’s own nesting. The reference list also carries Galí’s 1989 Columbia mimeo on keeping up with the Joneses and consumption externalities, Lucas (1978) on asset prices in an exchange economy, and Mehra and Prescott (1985).