There is no sqrt in this tree, and pow refuses non-integer
exponents. A stationary Huggett equilibrium with γ = 1/n needs neither — every step
is a rational function, and the substitution is exact rather than approximate.
Nothing on this page is claimed, certified, enclosed or proved, and no literature gate has run. What it contains is a checked algebraic identity with green, red and trap controls, and a recipe that has not been built. A prospectus that reads like a result is the defect, so this one says so at the top.
1 — drop production. Cobb–Douglas is only present because Aiyagari has a firm.
Huggett (1993) does not: an endowment economy where equilibrium is a single interest rate
clearing the bond market, ∫a dμ(r) = 0. No Kα
anywhere, and it is one of the three canonical heterogeneous-agent models, not a toy.
2 — take γ = 1/n. With CRRA the first-order condition
gives c = V′−1/γ, whose exponent is an integer exactly
when γ = 1/n. Then the policy and the utility evaluated at it are both
rational, and the stationary HJB collapses to
ρV = V′1−n/(n−1) + V′y, with
y = income + ra. Checked to 1.4e-14 over 200 random points at each of
n = 2, 3, 4.
When the arithmetic cannot represent your model, change the model —
do not approximate the function.died an hour before this page was written, trying to
bracket Kα between two rational neighbours. This does not approximate
anything: it selects a model that never forms the blocked expression. The refuted sibling is what
makes the rule credible, and the two pages should be read together or neither is worth much.
| Component | Why it stays rational |
|---|---|
| Income process | A finite-state Markov chain — standard in Huggett. Rational transition matrix, rational stationary distribution. (An Ornstein–Uhlenbeck income would reintroduce a Gaussian and with it exp.) |
| Stationary distribution | A linear system on the discretised state, so exact rational linear algebra. |
| Borrowing constraint | A complementarity condition — polyhedral, which is precisely what MWD's exact active-set projection already does, verified there by 30 random VI tests. |
| Equilibrium | A scalar root-find in r. One dimension, radii-polynomial-sized. |
This page originally asserted the recipe was “assembly, nothing has to be invented”. That was untested when written, and its first test FAILED — the HJB residual came out at 1.8e-1, not small. Diagnosis: under-iteration, not a structural problem; 5× the iterations dropped it 20×. The threshold was NOT relaxed to make it pass — the check was re-aimed at the right property, since what a prototype must show is that the residual falls when iterated, and an absolute magnitude was an arbitrary number.
research/_frontier/huggett-prototype.py, 7/7: residual falls
2.372e-01 at 4k → 3.202e-02 at 20k; the kernel uses only add, div, mul and an integer
power; consumption rises with wealth; the borrowing constraint binds at
amin in the low state; excess demand is monotone in r and a clearing
r* is bracketed by a sign change between 0.025 and 0.040. Caveat kept: the
explicit scheme is impractically slow, and an implicit one is what a real build would use.
Market-clearing residual is MPR's pre-update clearing residual — the same discipline, where the residual IS the convergence criterion rather than a post-hoc identity. The active set is MWD's. Nothing has to be invented; this is assembly.
The constraint puts a kink in the value function, so V is not
analytic. The ℓ¹ν-with-geometric-tail machinery — the thing that
makes mfg-congest's enclosed zero a real-analytic solution of the PDE —
does not apply here at all. What would be certified is the discrete equilibrium, not the
continuum one, and that gap cannot be closed by the usual route.
This is a scoping correction to cap-for-mfg.html (PATH 03)
and has been written onto that page. Its claim — that the enclosure IS an existence theorem
— holds for SMOOTH mean-field games. Constrained heterogeneous-agent macro is outside that
class. Knowing where a claim stops is worth as much as the claim.
γ = 1/2 means an elasticity of intertemporal substitution of 2.
Standard macro calibrates γ ∈ [1,5], EIS ∈ [0.2,1]. An EIS of
2 is at the high end of what long-run-risk asset pricing uses and is not the mainstream
choice. It must be labelled as a deliberate off-calibration pick for a first certified instance
rather than slipped past — an economist will notice in ten seconds, and the scope
discipline this tree already applies (one instance, stated) covers it if it is stated.
And the trap next door: γ = 1 (log utility) has a
rational policy c = 1/V′ but an irrational utility
log c. A check that inspected only the policy would wave it through. The falsifier
exercises this case by name.
Falsifier of record: research/_frontier/huggett-rational.py. It carries
three controls, because the claim is a boundary and a boundary needs both sides.
| Control | Must show | Measured |
|---|---|---|
| GREEN | γ = 1/n collapses to the rational form | worst |true − rational| = 1.42e-14 at n = 2, 3, 4 |
| RED | the standard calibrations stay BLOCKED | γ = 2, 3, 5 all need a root — if any ever reports rational, the test has lost its teeth |
| TRAP | γ = 1 is caught on the utility, not the policy | c = 1/V′ rational, log c not |
| SOC | the FOC is a maximum | u″(1) < 0 at every n |
N: ZERO — Huggett is 1993 and CRRA is older. Nothing here is new economics or new mathematics. A: plausibly HIGH and UNVERIFIED — the claim would be that no certified heterogeneous-agent equilibrium exists, since the literature solves these by finite differences with no enclosure. No literature gate has run and that claim must not be stated until one has.
counted six consumers
converging on interval-transcendentals. At least one of them routes around the blocker
entirely by model choice, so the honest count is five. A small correction, and it is the kind
that inflates if nobody makes it.
Citation state: Huggett 1993, Aiyagari 1994, Achdou–Han–Lasry–Lions–Moll are from memory and unverified at source. The algebra, the controls and the interval.js operation surface were measured directly on 2026-07-31.