Ramsey Meets Laibson in the Neoclassical Growth Model

The paper’s starting point is that most macro frameworks assume a constant rate of time preference for reasons that are “unclear,” while Laibson’s evidence shows people are “highly impatient about consuming between today and tomorrow but…much more patient” about choices further out, which creates a Strotz-type time-consistency problem because “the relative valuation of utility flows at different dates changes as the planning date evolves.” (Introduction, pp. 1125-1126). Barro frames the paper as asking “whether economists can continue to rely on the neoclassical growth model as their workhorse framework,” and states the paper “largely answers this question in the affirmative” while also extending the model’s empirical content (p. 1126).

Utility from date τ’s perspective is modified from the standard U(τ) = ∫u[c(t)]·e^(-ρ(t-τ))dt to U(τ) = ∫u[c(t)]·e^(-[ρ(t-τ)+φ(t-τ)])dt, where φ(·) is a function of only the time distance (t-τ), normalized to φ(0)=0, with φ′(v) ≥ 0 and φ′(v) → 0 as v → ∞ (Section I, eq. 1-2, pp. 1126-1127). This means the instantaneous rate of time preference, ρ+φ′(t-τ), “is high in the near term but roughly constant at the lower value ρ in the distant future” (p. 1127) — exactly the short-run-impatience, long-run-patience pattern Laibson [1997a] documents.

Under full commitment the household’s first-order condition becomes the usual Ramsey formula with ρ replaced by ρ+φ′(t-τ), so time-preference rates start high and decline toward ρ, making the steady state coincide with the standard model’s (Section II, eq. 5, pp. 1128-1129). Barro notes a conceptual problem: the current date τ at which commitment is assumed to begin “is arbitrary,” and if perpetual commitment were genuinely feasible it would likely have existed since “the infinite past,” in which case φ′(t-τ) would be zero for all t≥0 and the standard Ramsey result would hold throughout, not just asymptotically (p. 1128-1129). This motivates solving the no-commitment case, which Barro calls “the more basic problem.”

Barro derives that without commitment, log-utility consumption is still a constant fraction λ of wealth (capital plus the present value of wages), where λ = 1 / ∫₀^∞ e^(-[ρv+φ(v)])dv — a constant that can equivalently be written as a time-invariant weighted average of the instantaneous rates of time preference, ρ+φ′(v), and that satisfies ρ ≤ λ ≤ ρ+φ′(0) (Section III, eq. 10, 13, 14, pp. 1130-1133). Because the income and substitution effects from future interest rates cancel under log utility just as in the standard Ramsey model, “the income and substitution effects associated with future interest rates would still cancel under log utility, even though the rate of time preference is variable,” so the conjecture that consumption is a constant fraction of wealth “turns out to be correct” (p. 1130).

Because the effective rate of time preference λ is constant under log utility, “the model with log utility and no commitment is observationally equivalent to the conventional neoclassical growth model,” since the equilibrium coincides with the standard model’s “for a suitable choice of ρ” (Section III, p. 1134). Barro notes the practical implication: “since the parameter ρ cannot be observed directly, there is a problem in inferring from data whether the instantaneous rate of time preference includes the nonconstant term φ′(v)” — identification would require comparing populations or societies with differing commitment technologies rather than time-series data on a single economy (p. 1134-1135).

The short-run and long-run effects of introducing full commitment run in opposite directions: “the short-run impact of introducing perpetual commitment is…opposite to the long-run effect” — on introduction, the rate of time preference jumps up from the no-commitment average λ to the high short-run rate φ′(0), then declines asymptotically to the lower long-run rate ρ, which is why “the full commitment regime has lower ρ and higher k and c*” in the long run** (Section V.1, pp. 1138-1139). Table I’s numerical simulations (Cobb-Douglas production, ρ=0.02, δ=0.05, n=0.01, x=0.02, decay rate γ=0.50) show that when the short-run instantaneous discount rate φ′(0)=β is very high (0.50 per year), full commitment raises steady-state capital 96 percent and consumption 26 percent above the no-commitment levels, and the compensating variation for commitment is 26 percent of the initial capital stock — but for more moderate short-run impatience (β=0.20 or 0.10), these effects and the compensating variation shrink to a few percent or “essentially nil” (Section V.1, Table I, pp. 1139-1144).

Partial commitment lets households commit consumption over a rolling horizon of length T rather than committing forever (T=∞) or not at all (T=0); the resulting propensity to consume out of wealth, λ_T, “declines monotonically from λ₀ to ρ as T rises from zero to infinity,” so better commitment technology (higher T) means lower effective time preference and higher saving and capital in the long run (Section V.2, eq. 25-26, pp. 1144-1145). Quantitatively, under the paper’s baseline decay parameter (γ=0.50 per year), “a value T=5 is sufficient to get close to the T=∞ results” — five years of commitment ability is “nearly equivalent to the ability to commit forever,” while even T=1 year captures a substantial fraction of the value of full commitment (19 percent compensating variation versus 26 percent for full commitment, in the high-β case) (Section V.2, Table II, pp. 1145-1147).

No — Barro shows that “the steady-state interest rate under a general concave utility function coincides with that for log utility,” i.e., the same formula (eq. 28) for ρ holds regardless of the curvature of utility* (Section VI, eq. 28, p. 1148). However, the transitional dynamics change: under isoelastic utility with parameter θ, the time-varying effective discount rate λ(t) becomes a weighted average of future instantaneous time-preference rates with weights that depend on the path of interest rates relative to future consumption growth, and “if θ>1…the net effects from interest rates are opposite to those” that hold when θ<1 (Section VII, eq. 32-33, pp. 1148-1150). The precise transition path, Barro notes, is difficult to characterize analytically and “would probably be feasible” only via numerical simulation (p. 1150).

Barro surveys mechanisms that plausibly govern commitment ability, including credit-market features (pension-withdrawal penalties, the use of future labor income as loan collateral, bankruptcy law, mortgage-market development, usury laws), legal devices (estate-breakup restrictions, prohibitions on gambling/drugs/prostitution), advance-planning requirements (travel and restaurant reservations), durable goods, and “personal discipline and self-control,” including religiosity “measured by church attendance or outlays on organized religion” (Section V, pp. 1135-1138). Because the model implies that societies or families with better commitment technology (higher T) have lower effective time preference and higher saving in the long run but lower saving when commitment newly improves (the short-run effect is reversed), the paper’s “more novel empirical implications relate to the dynamic effects from an institutional change,” since a new commitment device “first lowers saving and capital accumulation” before raising it only “after a significant delay” (Section V.1, p. 1142).