Buffer-Stock Saving and the Life Cycle/Permanent Income Hypothesis

Carroll opens by noting that in the Federal Reserve’s 1983 Survey of Consumer Finances, 43 percent of consumers named “emergencies” as their most important saving motive, versus only 15 percent who named retirement – an answer the standard LC/PIH model does not obviously predict (Introduction, p. 1). He argues this and other evidence are consistent with a version of the LC/PIH model in which consumers face important income uncertainty but are both “prudent,” in Miles Kimball’s [1990b] sense of having a positive third derivative of utility (a precautionary saving motive), and “impatient,” in the sense that “if future income were known with certainty they would choose to consume more than their current income” (p. 1). Under these joint conditions, consumers engage in what Carroll calls “buffer-stock” saving: they have a target wealth-to-permanent-income ratio such that wealth below target triggers saving (prudence dominates) and wealth above target triggers dissaving (impatience dominates) (p. 1).

No – the utility function must also exhibit Decreasing Absolute Prudence, a property satisfied by the Constant Relative Risk Aversion (CRRA) utility function used throughout the paper (footnote 3, p. 1, citing Kimball [1990a, 1990b] for the argument that Decreasing Absolute Prudence is a natural condition to impose on utility functions). The existence and stability of the target wealth-to-income ratio is proved in a companion paper, Carroll [1996].

Even with a fixed aggregate interest rate, if consumers are sufficiently impatient, average consumption growth will equal average labor-income growth – for individual households and for aggregate consumption alike – even though these same consumers obey the standard consumption Euler equation that has “been widely thought to imply that consumption growth depends only on tastes” (Introduction, pp. 1-2). Carroll traces this result to a modeling shortcut in prior work: the log-linearized Euler equation contains a second-order variance term that most research had assumed was safely ignorable, constant, or zero, but which is in fact “an endogenous equilibrating variable” that on average takes on whatever value is required to make consumption growth track income growth (p. 2).

The condition is that, if future income were certain, consumption growth over the remainder of life would be slower than income growth – equivalently, that the consumer would want to dissave or borrow today to finance current consumption (Section III.A, pp. 9-10). Carroll is explicit that “this equation can be satisfied by consumers who do not discount future utility at all (ρ = 0) but who face positive income growth” (p. 10) – so impatience in his technical sense need not come from a literal high subjective discount rate; sufficiently rapid expected income growth alone can satisfy it.

The optimal consumption rule is strictly concave (proved in Carroll and Kimball [1996] for a wide class of problems including this one), so the marginal propensity to consume is a strictly decreasing function of wealth; poorer consumers therefore experience larger consumption swings from a given amount of income variation than wealthier consumers with a lower MPC (Section III.A, pp. 11-12). This is why the gap between expected consumption growth and the certainty-model growth rate – and, correspondingly, the variance of consumption growth – declines sharply as the wealth-to-income ratio rises (p. 12), and why, at the target wealth ratio itself, expected consumption growth is strictly less than expected permanent-income growth, a correction to an “approximately equal” claim in Carroll’s own earlier [1992] paper (equation 7, pp. 12-13).

In steady state, aggregate consumption growth converges to aggregate labor-income growth (equation 10) through the same endogenous-variance mechanism that equates growth rates at the household level, and this convergence is fast: about a two-year transition half-life under baseline parameters, versus roughly fifteen to twenty years (or longer) in a standard Solow or Cass-Koopmans growth model (Section III.D, pp. 22-23). Carroll argues this speed is what let the buffer-stock model, rather than a coincidental long-run balanced-growth steady state, actually explain the Carroll and Summers [1991] finding that consumption tracks income over three-to-five-year (not multi-decade) horizons, and also explains their finding of no cross-country relationship between aggregate consumption growth and average interest rates, since the omitted, theoretically interest-rate-correlated variance term is not observed in aggregate data (pp. 23-24).

Carroll argues that “typical methods of Euler equation estimation… yield meaningless results if the consumers involved are buffer-stock savers,” because those methods assume the Euler equation’s variance term is zero or constant (Section III.D, p. 19). He works through two examples: (1) Lawrance’s [1991] finding that more-educated households have faster consumption growth, which she attributes to a lower discount rate, can instead arise purely from an omitted-variable problem if education is correlated with income growth (pp. 19-20); and (2) Dynan’s [1993] near-zero coefficient estimates on both the interest-rate and variance terms, which she interprets as evidence against a precautionary motive, can instead arise precisely because buffer-stock consumers in different groups end up with the same consumption growth rate (their shared income growth rate) despite systematically different wealth, interest rates, and variance terms – so her finding is “actually supportive of a buffer-stock model,” not evidence against one (pp. 20-21). Consistent estimation across groups is possible only under fairly stringent conditions: important, predictable cross-group variation in both interest rates and income growth, and no cross-group variation in tastes (p. 21); the one class of consumers for whom standard Euler-equation methods remain reliable is effectively infinitely wealthy consumers (p. 21).

Across occupational groups, differences in age/income profiles are closely paralleled by differences in age/consumption profiles over periods of a few years or longer (Section IV.A, pp. 28-29, citing Carroll and Summers [1991] Figure IV). The unconstrained standard LC/PIH model has no explanation for this, since consumption growth there depends only on tastes; a simple Keynesian consumption function (C = α0 + α1Y) can explain it if α0 is small and α1 near one, but the Campbell-Mankiw [1989] hybrid model (α = 0.5) predicts a parallel that is “simply too close” to the data (p. 29). Simulating the finite-horizon buffer-stock model with three occupation-calibrated age/income profiles (Unskilled Labor, Operatives, Managers), Carroll finds consumption growth closely parallels income growth until roughly age 45-50, after which some retirement saving begins (Figure V, p. 29).

Household-level data show consumption is often far from current income even while aggregate consumption tracks aggregate income; Carroll follows Friedman’s [1957] errors-in-variables logic that regressing observed consumption on observed (rather than permanent) income biases the estimated slope toward zero (Section IV.B, pp. 31-32). He derives (equation 11) that under the permanent-income model the variance of the consumption/income ratio should move one-for-one with the variance of transitory income shocks across groups, while the simple Keynesian model predicts no such relationship. Using 1960-1961 Consumer Expenditure Survey data by occupation together with Carroll and Samwick’s [1995a] transitory-variance estimates, a regression yields var(C/Y) = .651·var(v/P) − .0072 (standard errors .128 and .0059), implying an MPC out of transitory income of about 0.2, statistically distinguishable from both zero and one at the 5 percent level (p. 34) – evidence Carroll characterizes as “stronger evidence against the Keynesian model than against the standard LC/PIH model” because measurement-error bias works against finding this result (pp. 34-35).

Diamond and Hausman [1984] and Table V of the paper (1963, 1983, 1989 Surveys of Consumer Finances) show the median household holds financial-asset-to-income ratios between 2 and 35 percent at all working ages, rising only modestly until shortly before retirement, and this pattern is stable across three decades despite a sharp post-1973 productivity growth slowdown (Section IV.C, pp. 35-36). Carroll shows the standard LC/PIH model, calibrated (at an 8 percent interest rate) to roughly match 1960s wealth data, implies that a plausible income-growth slowdown should have raised household wealth/income ratios by “roughly two years’ worth of income” on average – “more than an order of magnitude greater than the actual increase” (Figure VI, pp. 36-37) – whereas the buffer-stock model’s age/wealth profile shifts only slightly under the same experiment (Figure VII, p. 37), consistent with the modest actual rise observed between the 1963 and 1983/1989 surveys.

Table I shows the average MPC out of transitory income for buffer-stock consumers is at least 15 percent, and up to 50 percent, across the parameter values considered – versus a maximum of about 8 percent under any commonly used standard-model parameterization (2 percent at baseline values) (Section III.E, p. 25). By contrast, the MPC out of anticipated future (“human”) income is very small in the buffer-stock model (versus 3.8 percent in the standard model at the chosen parameters) and, expressed as an “implied discount rate” for future income (Appendix III), reaches “13,981 percent” at a low gross wealth ratio of 0.2, and roughly 22 percent even at the model’s target wealth ratio of about 1.6 (Section III.E, pp. 26-27) – a magnitude Carroll connects to Friedman’s [1963] own statement that households appear to discount future income at “33 1/3 percent.”

Carroll is explicit that the model is not a complete theory of household wealth: it substantially underpredicts the wealth held by the richest households (who held 64 percent of directly-held financial assets in 1983), is a poor vehicle for housing investment given its single perfectly liquid, riskless asset, and implies wealth falls to zero exactly at a certain, known date of death – an artifact of assuming no bequest motive and no uncertainty (e.g., medical expenses) beyond labor income (Sections IV.C and VI, pp. 38-39, 42-43). He suggests the model is best understood as describing the “truly discretionary ‘high frequency’ saving decisions of the median consumer” after pension and mortgage commitments are set aside, not as a full account of retirement saving, housing, or the behavior of the wealthy (p. 42).

Carroll argues that, read carefully, Friedman [1957] anticipated much of the buffer-stock reasoning, quoting Friedman’s caution that it “would… be a serious mistake” to equate permanent income with average lifetime earnings, since permanent income depends on a household’s own horizon (p. 30, quoting Friedman pp. 22-23, 93). He also notes Friedman [1963] separately estimated a marginal propensity to consume out of purely transitory income of about 0.3, a magnitude the buffer-stock model (with average MPCs of 15-50 percent) can match but the standard LC/PIH model cannot (Section IV.B, p. 35).