Relative risk aversion at most one #
Numbered Light references use the October 31, 2018 author draft, printed page 13: https://www.researchgate.net/publication/314238962
Light (2018) proves that individual saving rises with the interest rate — his Theorem 1, the
household half of the uniqueness argument — under the assumption that u'(c) · c is
non-decreasing. With u' > 0 that is exactly relative risk aversion -c u''/u' ≤ 1, because
d/dc [c u'(c)] = u'(c) · (1 - RRA(c)).
Economically it is the condition under which the substitution effect of a higher interest rate
dominates the income effect. It is the boundary of the region where the model is known to have
a unique equilibrium: Açıkgöz (2018) exhibits multiplicity at γ = 6.5, and Walsh and Young
(2025) report never seeing multiplicity below about γ = 1.49.
Stating it without derivatives #
u'(c) · c is the derivative of t ↦ u(eᵗ), so the assumption says precisely that u is
convex as a function of log consumption, and that is how it is defined here. The definition
needs no differentiability, and neither does its main consequence: increments of u across a
fixed PROPORTIONAL change in consumption grow as consumption grows,
u(α c₁) - u(c₁) ≤ u(α c₂) - u(c₂) for 0 < c₁ ≤ c₂ and α ≥ 1,
which is sub_le_sub_of_scale below. This is the multiplicative twin of
ConcaveOn.sub_le_sub_of_shift, and it is the form the comparative-statics argument uses, since
a change in the interest rate scales the return on saving rather than shifting it.
Where derivatives are available the two formulations agree, and relativeRiskAversionLeOne_iff
proves that. The examples at the end pin the boundary down: CRRA satisfies the condition exactly
when γ ≤ 1, log utility sits on the boundary (convex in log consumption because it is AFFINE
there), and for γ > 1 it fails — indeed u ∘ exp is then strictly concave, so the condition
fails as badly as it can.
Relative risk aversion at most one, stated without derivatives: u is convex as a
function of log consumption. Where u is differentiable with u' > 0 this is equivalent to
-c u''(c) / u'(c) ≤ 1; see relativeRiskAversionLeOne_iff.
Equations
Instances For
The derivative-free content of u'(c) · c increasing. The gain from a fixed
PROPORTIONAL rise in consumption is larger at higher consumption.
This is the multiplicative counterpart of ConcaveOn.sub_le_sub_of_shift, and it is the form
the comparative statics uses: a higher interest rate scales the payoff to saving.
The equivalence with u'(c) · c increasing #
Where u is differentiable on (0, ∞), t ↦ u(eᵗ) is differentiable with derivative
eᵗ u'(eᵗ) — which is u'(c) · c read at c = eᵗ.
If u'(c) · c is non-decreasing then relative risk aversion is at most one.
Conversely, relative risk aversion at most one makes u'(c) · c non-decreasing.
Relative risk aversion at most one is exactly u'(c) · c non-decreasing.
Where the boundary is #
CRRA utility in log consumption is eᵏᵗ / k with k = 1 - γ, so everything below turns on the
sign of k.
CRRA with γ < 1 has relative risk aversion at most one. In log consumption the
utility is e^{(1-γ)t} / (1-γ), whose derivative e^{(1-γ)t} rises because 1 - γ > 0.
Log utility sits exactly on the boundary: in log consumption it is the identity, which
is affine, hence convex — but only barely, since it is concave too. Log utility is γ = 1.
CRRA with γ > 1 fails, and fails strictly. In log consumption the utility is
e^{(1-γ)t} / (1-γ) with a negative exponent and a negative divisor, whose derivative
e^{(1-γ)t} strictly falls, so it is strictly CONCAVE in log consumption.
This is the reason the uniqueness literature stops at γ ≤ 1: beyond it the household's
response to the interest rate is not signed by this argument, and Açıkgöz (2018) and Walsh and
Young (2025) find calibrations where capital supply genuinely bends back.
The step of Light's Theorem 1 that all of this serves #
Light's Theorem 1 compares a household facing gross return α₁ with one facing α₂ ≥ α₁, and
its pivotal inequality is that the marginal value of wealth, α · u'(c), is larger at the
higher return. Written out, α u'(c) = (α / c) · (u'(c) · c), and the two factors are exactly
the two conditions: α / c(α x) rises with α because the consumption function is CONCAVE
(Lemma 4, ConcaveOn.div_le_div_of_scale), and u'(c) c rises with c because relative risk
aversion is at most one. Neither factor alone signs the product; splitting it this way is the
whole trick.
Note what the consumption function has to supply: concavity, positivity at zero assets, and
monotonicity. The first is Light's Lemma 5 and is the deep one — he cites Jensen (2017), and it
is a Carroll–Kimball result that needs more than concavity of u and of the value function.
The pivotal inequality of Light (2018) Theorem 1. For a concave, positive, increasing
consumption function c and marginal utility du with relative risk aversion at most one, the
marginal value of wealth α · u'(c(α x)) is non-decreasing in the gross return α.
The pivotal inequality on a bounded asset interval, with the asset rescaled by the ratio of two positive gross returns.
Beyond relative risk aversion one #
For u'(c) = c^{-γ} with γ > 1, c · u'(c) = c^{1-γ} FALLS, so the second factor above goes
the wrong way — but only by the factor (c(α₂x)/c(α₁x))^{γ-1}, and the first factor bounds that
ratio: concavity through a positive value at zero gives the chord bound
c(α₂x)/c(α₁x) ≤ 1 + (1 - c(0)/c(α₁x))(α₂/α₁ - 1). So the pivotal inequality survives wherever
(1 - c(0)/c(α₁x)) · (ρ - 1) ≤ ρ^{1/γ} - 1, ρ = α₂/α₁,
which for returns close together says consumption at α₁x is at most about γ/(γ-1) times
consumption at zero assets. For γ ≤ 1 the condition is automatic, since ρ^{1/γ} ≥ ρ;
for γ > 1 it is a restriction on the RANGE of the consumption function, not on the utility.
Light's pivotal inequality beyond relative risk aversion one, for u'(c) = c^{-γ} and
any γ > 0: α₁ · c(α₁x)^{-γ} ≤ α₂ · c(α₂x)^{-γ} for α₂ = ρ α₁ ≥ α₁, provided
(1 - c(0)/c(α₁x))(ρ - 1) ≤ ρ^{1/γ} - 1.