A natural asset bound for log utility #
exists_decline_of_consumption_lower_bound turns a LINEAR lower bound on consumption into the
statement that assets above an explicit threshold decline — the natural asset bound, which is
what assetCap has been standing in for, and what
not_concaveOn_consumptionFn_of_cap_binds says has to hold before the Carroll–Kimball
hypothesis is even true. Its input, the linear bound, has been missing. This file supplies it
for log utility.
The argument is the crudest possible deviation #
Saving nothing is always feasible. Comparing the optimum with it,
u (resources) + β · cont 0 ≤ u (consumption) + β · cont (policy),
so u resources - u consumption ≤ β (cont (policy) - cont 0) ≤ 2 β ‖V‖. For LOG utility that
reads log (resources / consumption) ≤ 2 β ‖V‖, which is a linear bound outright:
consumption ≥ exp (-2 β ‖V‖) · resources.
Log and the other CRRA regimes #
For log utility, u resources grows without limit and the deviation gives the proportional
consumption bound above. In the unbounded-domain CRRA branch with γ > 1, utility is bounded
above, so this comparison gives a constant consumption floor rather than that linear bound.
The separate bounded-domain treatment also permits 0 < γ < 1; crra_pinch does not exclude
that regime from the shared household structure.
What is honest about the constant #
exp (-2 β ‖V‖) is very lossy. The sharp constant is Ma and Toda's asymptotic MPC
c_bar = 1 - (β R^(1-γ))^(1/γ), at which the decline condition becomes exactly β R < 1
(one_sub_mpc_mul_of_asymptotic). Getting it needs the Euler equation, hence the envelope
condition, hence the machinery that IncomeFluctuationEnvelope supplies only under side
conditions. So the bound proved here is qualitatively right and quantitatively weak: it
establishes a natural asset bound, but only for interest rates small relative to exp (-2 β ‖V‖),
where the sharp argument would ask only for impatience.
The slack in the "save nothing" comparison: twice the size of the value function, discounted.
Equations
- P.deviationGap = 2 * ↑P.discount * ‖P.toExtended.valueFunction‖
Instances For
The optimum beats saving nothing, so utility cannot fall far short of the utility of eating everything. This holds for any utility function; it is the log case that turns it into a linear bound.
A linear lower bound on consumption, for log utility. The one place the development gets a bound that scales with resources rather than being a constant.
The natural asset bound for log utility. Assets above an explicit threshold decline, so the household never accumulates without limit and the asset cap can be justified rather than imposed.
The hypothesis on the interest rate is the price of the crude constant: the sharp argument would
ask only for β (1 + r) < 1.
A better constant, from a smaller deviation #
log_consumption_linear_lower_bound compares the optimum with saving NOTHING, which is a huge
deviation and costs an exponential: ε = exp (-2 β ‖V‖). Comparing instead with saving a
FRACTION θ of the optimum costs only a ratio, and letting θ approach one turns the bound
polynomial:
consumption ≥ resources / (1 + 2 · 2 β ‖V‖).
No derivative is taken. The reason a near-zero deviation is affordable is that BOTH sides shrink
with 1 - θ: the utility gain is log (1 + (1-θ) b / c), and the continuation loss is bounded by
2‖V‖ (1-θ) / θ because concavity makes the slope of cont on [θb, b] no larger than its slope
on [0, θb]. The factor (1-θ) cancels, and what survives is the comparison of the two
constants.
θ = (G + 1/2) / (1 + G) is chosen so that θ (1 + G) - G = 1/2 exactly, which is what makes the
final inequality b ≤ 2 G c.
The gain is small when ‖V‖ is small and large when it is not: at G = 1.3 the two constants are
0.27 and 0.28, at G = 5 they are 0.007 and 0.09. Since ‖V‖ grows like |log minIncome|,
it is the dispersed calibrations — the ones the precautionary motive needs — where this matters.
A polynomial linear lower bound on consumption. Sharper than
log_consumption_linear_lower_bound whenever the value function is large, and proved from a
deviation to θ times the optimal saving rather than to zero.
The other half of exhaustion: the corner, for log #
exists_decline_of_consumption_lower_bound gives assets falling above a threshold. The ergodic
argument also needs the constraint to BIND below one — policy = 0 on [0, a₀] at the bad income
state — and policy_eq_zero_of_corner_at supplies that from two explicit constants: a Lipschitz
bound L on the value function and a lower bound m on the secant slope of u, with β L < m.
Both are computable for log. The secant slope of log on (0, R] is at least 1/R, from
log x ≤ x - 1. The Lipschitz constant is the fixed point of the operator's own estimate,
L = K (1+r) / (1 - β(1+r)) with K = slopeBound log minIncome = 2 log 2 / minIncome, so β L < m
reads
resources · β · 2 log 2 (1+r) / (minIncome (1 - β(1+r))) < 1,
a condition on the state's resources alone. Since resources at the bad state are
minIncome + (1+r) a, it cuts out an interval of low assets — exactly what the ergodic argument
wants, and exactly why the condition has to be state-dependent rather than global.
The slope bound of log at the income floor, in closed form.
The Lipschitz constant the operator's own estimate is a fixed point of.
Equations
Instances For
The value function is Lipschitz with the explicit log constant.
The borrowing constraint binds where resources are small, with an explicit threshold in the primitives.
The bound is not vacuous. At r = 0 the interest-rate hypothesis holds automatically,
since exp is positive, so the log witness has a natural asset bound outright.