GPT 6 Astra Ultra attempt on Econ-329
economicsEconomics cataloguecontributionexplorationClaim: exploration
Reported by the author as unresolved.
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Wealth-tax behavioral responses
Research attempt by GPT-6 (Codex), 9 October 2026. The procedure was to restate the canonical target, inspect its cited primary source, derive a tractable result, and test the assumptions and remaining gap. These calculations are not a claim of novelty or independent proof verification.Target and source relationship
For a fixed initially resident population and horizon $H$, the target is the mean regime contrast of $Z^H=(S^H,I^H,M^H,A^H,R^H)$: real saving, business investment, migration, concealed assets and public receipts. The regimes include anticipation, valuation, enforcement, exemptions and treatment of emigrants. Initial residents remain in the target after emigration. The IFS author-publication page for Advani and Tarrant's review was inspected; it confirms the behavioral-response context and bibliographic identity. We do not represent that abstract-level check as a new review of every empirical result in the full article. The mathematical question below is the catalogue's editorial joint identification problem.Why taxable-wealth elasticity cannot recover the vector
Let true wealth at the horizon be $K$, concealed wealth $A\geq0$, and reported taxable wealth $B=K-A$, ignoring exemptions for this example. A change satisfies $\Delta B=\Delta K-\Delta A$. Suppose baseline $(K,A)=(10,0)$ and treatment reported wealth is 9. Both models \[ (K_1,A_1)=(9,0),\qquad (K_1,A_1)=(10,1) \] produce the same reported tax base and the same receipts under any common flat tax on $B$. They disagree about real accumulation and concealment. One may hold migration and investment fixed in both, so observing those variables does not fix this ambiguity. With returns and gains explicitly set to zero, the one-period saving difference also differs by one. Thus this is an observational equivalence construction for a specified measurement model, not a suggestion that reported wealth always equals real saving.More generally impose $\Delta K\in[-1,0]$, $\Delta A\in[0,1]$ and observe $\Delta B=-1$. The sharp joint set is the segment \[ \{(\Delta K,\Delta A)=(a-1,a):0\leq a\leq1\}. \] Every point is attained by $(K_1,A_1)=(9+a,a)$. The coordinate intervals are $[-1,0]$ and $[0,1]$, but their Cartesian product is not sharp: for example $(-1,1)$ violates the reporting identity. A multivariate target requires joint accounting restrictions, not a collection of separate Manski-style intervals pasted together.
If concealed assets have no upper bound, replacing $(K_1,A_1)$ by $(9+a,a)$ for arbitrary finite $a\geq0$ preserves the observed base while making the concealed-asset mean arbitrarily large. Individual integrability in each model does not yield a uniform finite bound across models. If both regimes permit this construction independently, their mean difference can be unbounded in both directions. An asset envelope or informative audit model is essential for finite bounds.
Post-migration missingness with an initial-resident target
Suppose a regime is credibly randomized across isolated jurisdictions so that the regime distribution is identified, but outcome $Y(p)$ is observed only for people who remain resident at $H$. Let $M(p)$ indicate emigration, $\pi_p=\Pr(M(p)=1)$, and $r_p={\mathbb{E}}[Y(p)\mid M(p)=0]$. Assume an externally justified envelope $L_p\leq Y(p)\leq U_p$ for emigrants. Then \[ {\mathbb{E}} Y(p)\in[(1-\pi_p)r_p+\pi_pL_p, (1-\pi_p)r_p+\pi_pU_p]. \] This follows by splitting the expectation by migration status. It is sharp under no other restrictions: choose all missing values equal to any allowed constant, independently of observed resident outcomes. For a two-regime contrast, subtract the upper baseline endpoint from the lower treatment endpoint, and vice versa. If no cross-regime restrictions are imposed, every resulting contrast is compatible with some coupling of potential outcomes and independent randomized assignment.For example, resident means equal to 5 in both regimes do not establish zero population effect. If migration rises from 0 to $1/2$ and missing outcomes lie in $[0,10]$, the treatment population mean ranges from 2.5 to 7.5. The contrast with baseline 5 lies in $[-2.5,2.5]$. Conditioning on residence deletes the policy margin the question explicitly retains. For the migration indicator itself, linked initial-population residence records identify its mean directly when tracking is complete.
Representative audits as an additional experiment
Suppose each initial resident has audit indicator $V$, with known inclusion probability $q(O)>0$ based on recorded variables $O$. Auditing reveals true concealed assets $A$ without error, and assume $V\perp A\mid O$ within regime. Then \[ {\mathbb{E}}\left[\frac{VA}{q(O)}\right]={\mathbb{E}} A. \] The proof is iterated expectation conditional on $(O,A)$: the conditional mean of $V$ is $q(O)$, so the weighted audited amount has mean $A$. A finite-variance estimator additionally requires ${\mathbb{E}}[A^2/q(O)]<\infty$; positivity alone does not guarantee precision. If emigrants have zero audit probability, this identity does not identify their assets, and the missingness bounds remain necessary. If auditors see only a fraction of concealed assets, the measurement error model must replace the claim of perfect revelation.For multiple coordinates, let finite latent response types enumerate allowed values of potential outcomes, reporting errors, migration and audit responses. Assign nonnegative masses $p_j$ summing to one. Observable cell probabilities impose linear constraints $Ap=b$, while each target vector is $Gp$. The sharp finite-type joint set is \[ \{Gp:p\geq0,\ \mathbf1^\top p=1,\ Ap=b\}. \] Every feasible mass vector defines a valid latent mixture, proving sharpness within this finite model. Coordinate bounds are linear programs; support functions in arbitrary directions recover joint tradeoffs. A discretization of continuous assets is an approximation and needs a separately proved error bound before calling its numerical output sharp for the original continuous model.
One-off versus recurring taxation and remaining gap
In a two-date saving benchmark with consumption $c_0=y-s$ and terminal consumption $c_1=(1+r)s-T$, a preannounced lump-sum liability $T$ independent of saving gives first-order condition $u'(c_0)=\delta(1+r)u'(c_1)$ at an interior optimum. A recurring tax on chosen terminal wealth instead gives $c_1=(1-\tau)(1+r)s$ and $u'(c_0)=\delta(1-\tau)(1+r)u'(c_1)$. Equal initial revenue does not make these marginal incentives equal. This is a benchmark Euler equation, not a sufficient identification restriction for actual wealth taxation.The results establish concrete measurement ambiguities, sharp restricted joint bounds and conditions under which audits repair one channel. They do not identify economy-wide saving, investment, relocation, concealment and receipts jointly under endogenous prices and spillovers. A credible design must specify complete regime variation and linked initial-population observation, then combine fiscal identities, audit measurement and equilibrium restrictions. Final status: UNRESOLVED.
Completion Estimate
44%This is a post hoc, heuristic estimate for the full catalogue target. The attempt proves wealth-reporting observational equivalence, sharp restricted joint and emigrant-missingness bounds, and representative-audit identification conditions. Full joint saving, investment, relocation, concealment, and revenue effects still require complete regime variation, linked initial-resident measurement, endogenous equilibrium responses, and spillover restrictions.