Weighted power means with a negative exponent #
The analytic engine of Carroll and Kimball (1996). For CRRA utility u' c = c ^ (-γ) the
first-order condition of a consumption–saving problem reads
c = (β R) ^ (-1/γ) · (∑ π z' · c' z' ^ (-γ)) ^ (-1/γ),
so the consumption chosen against a given stock of end-of-period assets is a positive multiple
of the weighted power mean, with exponent -γ, of next period's consumptions. Carroll and
Kimball's induction therefore needs exactly one fact: a power mean with a negative exponent
is concave and nondecreasing, so it carries concavity of next period's consumption functions
to concavity of this period's.
The proof #
powerMean w p is positively homogeneous of degree one, so concavity is the statement that the
set {x > 0 : powerMean w p x ≥ 1} is convex. With p < 0 that set is {x > 0 : ∑ w x ^ p ≤ 1}
— a sublevel set of ∑ w x ^ p, which is convex because t ↦ t ^ p is. So the whole thing comes
down to convexity of a single real function of one variable, and the scaling is bookkeeping.
convexOn_rpow_of_neg supplies that one function: Mathlib has x ^ p convex for 1 ≤ p and
concave for 0 < p < 1, but not the negative case, and it is proved here from concavity of
log and convexity of exp.
Kimball's risk-tolerance aggregation theorem is the differential form of the same fact. This is the integrated form, and it needs no derivatives.
The power mean #
Normalising by the mean makes the weighted sum exactly one. This is the statement that
the level set {powerMean = 1} is where the scaling argument lands.
The gradient sum of a power mean is at most M / min x. The quantity
∑ w x^{-γ-1}, which is what a common shift costs when propagated through a power mean, is bounded
termwise by ∑ w x^{-γ} / min x.
This is worth recording as a WARNING rather than a tool. Normalised, it says the gradient sum
A = ∑ w (x/M)^{-γ-1} is at most M / min x — and M / min x is exactly the multiplier that
bounding the expectation by its smallest term produces. Measured in the overlapping-generations
life cycle, A runs from 1.04 to 1.19 while M / min x runs from 4.7 to 5.9, so this bound
discards the entire advantage of keeping the certainty equivalent. A usable bound on A has to see
the WEIGHTS — the far states carry little probability — and not merely the support.
The gradient sum is one plus the variance of the risk tilt. Write u i = (x i / M) ^ (-γ)
for the tilt. By the definition of the power mean its weighted mean is exactly one, and then the
gradient sum ∑ w u ^ (1 + 1/γ) is, at γ = 1 where the exponent is exactly two, one plus the
weighted variance of u.
This is the bound the minimum cannot give. sum_mul_rpow_pred_le sees only the support of x and
returns M / min x, measured at 4.7 to 5.9; the variance sees the WEIGHTS, and is measured at
0.041. For γ > 1 the exponent 1 + 1/γ falls below two and Lyapunov's inequality gives
∑ w u ^ (1+1/γ) ≤ (1 + Var)^{(γ+1)/(2γ)}, which is an identity at γ = 1 and loses accuracy
above it: measured 1.076 against a true 1.050 at γ = 1.5, and 1.26 against 1.08 at
γ = 3, against an allowance of about 1.06 throughout. So this route closes at logarithmic
utility and not above (WriteUps/WriteUpOLG/numerics/lyap.m).
The sharp moment bound. For weights summing to one, u ≥ 0 with ∑ w u = 1 and
1 < p < 2, ∑ w u^p ≤ (∑ w u²)^(p-1). It is Jensen for the concave x ↦ x^(p-1) under the
tilted weights w u, which sum to one by hypothesis, and it is attained by the two-point law on
{0, ∑ w u²}. With p = 1 + 1/γ it bounds the gradient sum of the two-rate chain by
(1 + Var)^(1/γ), against Lyapunov's (1 + Var)^((γ+1)/(2γ)).
A power mean with a negative exponent is concave.
Homogeneity turns concavity into the convexity of {x > 0 : ∑ w x ^ p ≤ 1}, and that is a
sublevel set of a convex function. This is Kimball's risk-tolerance aggregation in integrated
form: no derivative appears anywhere.
The Carroll–Kimball step. A negative power mean of concave positive functions is concave: concavity of next period's consumption functions carries to the consumption chosen against a given stock of end-of-period assets.
And nondecreasing, which the endogenous-gridpoint transfer also needs.