Self-Control and Saving for Retirement

The paper opens by arguing that mainstream economic models assume consumers are unusually good at both making saving decisions and sticking to them, and that both assumptions are “dubious, particularly in regard to saving for retirement.” It cites three types of evidence for a systematic gap between intentions and actions: the popularity of commitment devices such as Christmas clubs and automatic payroll deductions (“in 1995 American consumers deposited their holiday savings in roughly 10 million Christmas club accounts”); survey evidence that consumers say they prefer flat or rising consumption paths but “actually implement downward-sloping consumption paths” when not liquidity constrained; and direct survey evidence that most people report saving less than they think they should – a 1997 Public Agenda survey found 76 percent of respondents believe they should be saving more, and among those who feel they should already be saving seriously, only 6 percent report being “ahead” versus 55 percent “behind.”

The paper models self-control problems with a discrete-time “quasi-hyperbolic” discount function, {1, βδ, βδ², βδ³, …}, which mimics the declining discount rates of the generalized hyperbolic function used in the psychology literature while remaining analytically close to the standard exponential model. Because β is set below 1, the discount factor between the current and very next period (βδ) is lower than the discount factor applied between any two later periods (δ), so “an individual’s preferences at period 0 differ from the same individual’s preferences at period t”: self 0 would prefer self t to be patient about the trade-off between t and t+1, but self t, once it arrives, applies its own steeper short-run discounting and disagrees. The authors model this as “an intrapersonal game played among the consumer’s temporally situated selves,” following Strotz (1956), rather than as a single intertemporal optimization problem.

The authors assume consumers have “unlimited sophistication in problem solving” and correctly anticipate how their future selves will behave, rather than modeling them as “naifs” (following Akerlof and O’Donoghue and Rabin) who falsely believe future selves will honor the current self’s plans. They adopt perfect rationality “not because it necessarily aptly describes consumer behavior, but rather because it represents the starting point for all economic analysis,” and note there is no well-developed bounded-rationality alternative for the life-cycle saving problem. They flag that this choice is not neutral: “had we chosen the naif model, we would probably have found smaller effects of DC plans for hyperbolic consumers,” since naifs do not value a DC plan’s commitment properties if they believe their future selves will behave well regardless.

The model follows Engen, Gale, and Scholz’s simulation approach but with three distinguishing features: it adds a stochastic (rather than deterministic) bequest process, it includes two illiquid assets (a defined-benefit pension plus a partially illiquid DC plan) rather than one, and it gives consumers hyperbolic rather than exponential discount functions. Consumers are divided into three education groups (high school dropout, high school graduate, college graduate) with separate income, retirement-age, and bequest processes estimated from the PSID; labor income follows a stochastic process with persistence and transitory components estimated by GMM; the DC plan replicates 401(k) rules (a $10,000 contribution cap, employer matching, and a tax penalty of w percent on withdrawals before age 60); and the three state variables (liquid assets, DC plan assets, and labor income) make the consumer’s problem a numerically solved backward-induction game rather than a standard contraction-mapping dynamic program, because the hyperbolic objective function “does not inherit concavity from u” and policy functions can be discontinuous and nonmonotonic.

The benchmark calibration sets the coefficient of relative risk aversion ρ equal to 1 (log utility), based on estimation procedures that do not rely on linearized Euler equations, and sets β = 0.85 for hyperbolic consumers versus β = 1 (no extra short-run discounting) for exponential consumers, with δ chosen so the two economies accumulate identical preretirement wealth absent a DC plan. The authors also report results for ρ = 3 “to demonstrate that our results are highly sensitive to the choice of ρ” and to compare with Engen, Gale, and Scholz, who use that value. With ρ = 1, the model’s hyperbolic economy implies a measured elasticity of intertemporal substitution of 0.22, which the authors say is “consistent with the available empirical evidence,” though they acknowledge available experimental evidence on time preference would favor a much lower β (around 0.6) than the 0.85 they use to avoid numerical pathologies.

At first glance “the life-cycle choices of hyperbolic and exponential consumers are indistinguishable,” since the calibration is designed to equalize their preretirement wealth accumulation, but the paper identifies two regularities that do distinguish hyperbolic consumers and that also show up in Survey of Consumer Finances (SCF) data: hyperbolic consumers “are more likely to face binding liquidity constraints,” and they reproduce the “missing” precautionary-saving coefficient that Dynan (1993) found anomalously low or negative when estimating a second-order-expanded Euler equation on CEX data. The paper shows analytically that a first-order term omitted from Dynan’s regression – present only when β < 1 – biases her estimated coefficient downward in exactly the direction she found, and confirms this by running her regression on the model’s own simulated data.

If hyperbolic households could costlessly and perfectly bind all future selves to the exponential household’s optimal rule, average peak wealth accumulation would rise from $184,000 to $298,000 (62 percent higher), and the nineteen-year-old self would value that commitment as much as an immediate, tax-free wealth windfall equal to 36 percent of consumption at age twenty – for ρ = 1; the welfare effect falls to the equivalent of only 2 percent of consumption, and peak accumulation rises only 15 percent, when ρ = 3. The paper uses this to motivate why real DC plans, as a partial, imperfect commitment device, should matter much more to hyperbolic than exponential households, but only when risk aversion is low.

With ρ = 1, adding a 401(k)-style DC plan with a 50 percent employer match raises the steady-state net national saving rate from 4.14 percent to 7.63 percent (an 84.3 percent increase) with a 10 percent early-withdrawal penalty, and to 8.35 percent (101.7 percent) with a 50 percent penalty, for exponential households; the corresponding increases for hyperbolic households are 99.2 percent and 134.0 percent. Welfare effects are larger still and more skewed toward hyperbolic households: twenty-year-old hyperbolic high-school-graduate households need a wealth transfer equal to 70 percent of consumption to be as well off without the DC plan as with it (under 50 percent matching and a 10 percent penalty), versus 22 percent for otherwise-identical exponential households. When ρ is raised to 3, the saving-rate increases shrink to a range of 23.2-38.2 percent for both types of household, converging toward the 9.1-17.2 percent increase that Engen, Gale, and Scholz report using ρ = 3 in a similar but exponential-only model – which the authors read as confirming and extending that paper’s result rather than contradicting it.

The paper identifies several design features of actual 401(k) plans that operate as commitment devices independent of their tax benefits: automatic payroll deduction, which creates a delay of “a month or two” before a change in contributions takes effect – long enough to defeat a hyperbolic consumer’s typically much shorter-run temptation – and withdrawal restrictions before age 59½ that impose a 10 percent (or larger) penalty plus loss of future tax deferral. However, the authors note that the recent, widespread adoption of 401(k) loan provisions (90 percent of plans surveyed by Hewitt Associates allowed loans against the balance) “has undermined the effectiveness of these plans as a commitment device,” even though such loans are themselves fairly restrictive (capped at 50 percent of the vested balance or $50,000, and typically repaid within five years).

The authors flag that their chosen β = 0.85 is more conservative than most experimental evidence on time preference would support (closer to 0.6), that the model does not evaluate the transition dynamics from a no-DC-plan to a DC-plan steady state (where short- and long-run effects of a tax incentive can have “the opposite sign”), and that it compares a world with one endogenous commitment device (the DC plan) to a world with none, when in reality consumers may simply substitute a new DC plan for a preexisting commitment device. They also raise, without resolving, a conceptual puzzle for the sophisticated-hyperbolic framework itself: if illiquid assets are valuable commitment devices and consumers are sophisticated enough to want them, “why have banks not filled the gaps” by selling commitment devices explicitly, rather than consumers getting commitment only as a byproduct of tax-favored retirement accounts.