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Econ-388 · economics · Economics

Wealth taxation

LLM status Partially solved unverified

Statement

Original atlas identifier: P2; entry 42. Field: Public finance and taxation. Original tractability label: Hard.

Classification: Parameterized theoretical/empirical research agenda; not a certified unsolved theorem.

Definitions and assumptions

Fix dates $t=0,\ldots,T$, discount factor $\beta\in(0,1]$, and an initial distribution $F$ over household wealth portfolios, liquidity, entrepreneurial ability, and residence. A recurrent-wealth policy $p$ specifies asset-specific tax bases, exemption thresholds, rates, valuation rules, reporting obligations, and enforcement. Economic wealth $A_{it}$, assessed wealth $\widehat A_{it}$, and reported wealth $\widetilde A_{it}$ are distinct variables for household $i$. Each model $m$ specifies investment, entrepreneurship, valuation error, avoidance, migration, and a selected general equilibrium. Let $EV_i(p;m)$ be discounted monetary equivalent variation, including private compliance and relocation costs. Let $R_t(p;m)$ be revenue across all affected taxes and $C_t(p;m)$ administrative resource costs. Fix nonnegative welfare weights $w_i$ and public-fund value $\lambda$; impose finite discounted moments.

Formal research question

Editorial formulation: the mathematical target below is not a theorem quoted from the references.

For a legally feasible class $\mathcal P$ satisfying enforcement-capacity limits and a stated present-value public budget, identify the welfare--revenue frontier \[ \mathcal F_m=\left\{\left(\sum_{t=0}^{T}\beta^t[R_t(p;m)-C_t(p;m)],\ \mathbb E_m[w_iEV_i(p;m)]\right):p\in\mathcal P\right\}. \] For each required net-revenue target $\bar R$, compare the supremum of weighted equivalent variation over policies meeting that target. Identify which asset-specific elasticities can be transported across valuation and reporting regimes, and give bounds when investment, reassessment, concealment, and mobility cannot be separately recovered. A policy dominates another only relative to the same welfare weights, horizon, equilibrium selection, and revenue requirement; a universal positive answer is not asserted.

Interpretation and scope

“Severe” distortion must be replaced by an explicit welfare-loss or entrepreneurial-entry constraint chosen before estimation. Illiquid business wealth requires both valuation and financing rules; a low assessed value can reflect an exemption or measurement error rather than low economic wealth. Study migrants in origin and destination records to distinguish an actual international relocation from local tax-base reassignment. Specify whether entrepreneurs can use instalments, loss offsets, or deferrals, since those rules alter the borrowing constraint and investment response. Danish and Swiss institutional evidence can discipline a model, but extrapolation to a new tax base or national policy requires invariance assumptions. A useful empirical deliverable jointly measures net receipts, private burdens, asset prices, firm entry, and residence rather than treating reported-wealth growth as saving.

Source audit and status

[S1] is an explicitly selected resolution of the ambiguous Saez--Zucman author-year citation. [S2] and [S3] provide existing country-specific evidence. None certifies that every design on this frontier is unstudied or that a globally optimal design is currently unknown.

References

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  1. Emmanuel Saez and Gabriel Zucman (2019). Progressive Wealth Taxation. Brookings Papers on Economic Activity, Fall, 437--533. https://www.brookings.edu/articles/progressive-wealth-taxation/
    Locator and support limits: Paper and abstract: wealth-tax design and simulations, not direct identification of all long-run responses.
  2. Katrine Jakobsen, Kristian Jakobsen, Henrik Kleven, and Gabriel Zucman (2020). Wealth Taxation and Wealth Accumulation: Theory and Evidence from Denmark. The Quarterly Journal of Economics 135(1), 329--388. DOI: 10.1093/qje/qjz032. https://academic.oup.com/qje/article-abstract/135/1/329/5584349
    Locator and support limits: Abstract and article: Danish wealth-tax reforms and accumulation; institutional transport requires additional assumptions.
  3. Marius Brülhart, Jonathan Gruber, Matthias Krapf, and Kurt Schmidheiny (2022). Behavioral Responses to Wealth Taxes: Evidence from Switzerland. American Economic Journal: Economic Policy 14(4), 111--150. DOI: 10.1257/pol.20200258. https://www.aeaweb.org/articles?id=10.1257/pol.20200258
    Locator and support limits: Abstract: reported wealth, mobility, and housing-price responses in Swiss cantons; reported wealth is not real saving.

Common conventions: Source mathematical conventions

Unless an entry states otherwise, agent and firm sets are finite. State and action spaces are measurable, strategies and outcome maps are measurable, probability laws are specified, and expectations exist for the targets under discussion. Infinite-horizon discount factors lie in $(0,1)$ and discounted objectives are finite. Norms, tolerances and complexity input sizes must be specified for computational comparisons. Constants or primitives that appear locally are inputs, not empirical facts asserted by this catalogue.

Identification. A statistical model $m\in\mathcal M$ implies an observable law $P_m$ and target $\theta(m)$. At law $P$, its identified set is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M,\ P_m=P\}. \] Point identification holds when this set is a singleton, or equivalently when $P_{m_1}=P_{m_2}$ implies $\theta(m_1)=\theta(m_2)$ for all compatible models. Sharp bounds mean bounds attained, or approached when the boundary is not attained, by compatible models. Writing this set is a definition, not a solution: the substantive task is to establish informative restrictions and derive or estimate the set. An unrestricted-confounding model may give only trivial bounds.

Inference. Identification, estimability and valid uncertainty quantification are separate questions. Fix $\alpha\in(0,1)$. A confidence procedure $C_n$ over a declared law class $\mathcal P$ has asymptotic uniform coverage at level $1-\alpha$ when \[ \liminf_{n\to\infty}\inf_{P\in\mathcal P}\Pr_P\{\theta(P)\in C_n\}\geq1-\alpha. \] For set-identified targets the coverage event must be defined separately, for example coverage of the full identified set. If an entry uses identification-indexed models instead of uniquely indexed laws, the same coverage convention is applied over those models. Pointwise limits do not establish uniform validity, and asymptotic validity does not guarantee finite-sample precision. Sample sizes, dependence structures and weak-identification sequences are part of each application.

Counterfactuals. $Y_i(a)$ is a potential outcome under a specified intervention, including its timing, implementation and versions. It is not $Y_i$ conditional on voluntarily choosing $a$. A full intervention vector permits interference; shorthand scalar treatment notation does not silently impose no interference. Assignment, selection, attrition, anticipation and measurement must be specified. A claim about an instrument's local effect is not automatically a population policy effect.

Economic equilibrium. A counterfactual model includes best responses, constraints, feasibility, market clearing, state transitions and expectations consistent with the stated information. When several equilibria exist, either a declared selector is an input or the target is set-valued. Comparisons across policy regimes must use the stated selection convention. Reduced-form relationships are not presumed invariant after an equilibrium-changing intervention.

Optimization and welfare. Optimization is over the stated feasible set. If attainment is not established, $\sup$ or $\inf$ and $\varepsilon$-optimal choices replace an asserted maximizer or minimizer. For example, with value $V=\sup_{a\in\mathcal A}W(a)<\infty$, an $\varepsilon$-optimal set is $\{a\in\mathcal A:W(a)\geq V-\varepsilon\}$ for $\varepsilon>0$. Benefits, costs, risk and health or environmental outcomes can be aggregated only using declared units and conversion weights. Interpersonal utility weights, the treatment of fiscal transfers and foreign profits, discounting, and the behavioral welfare criterion are explicit inputs. A comparative-static derivative additionally requires existence and appropriate differentiability of the selected counterfactual.

Sources. Local labels [S1], [S2], and so forth restart in every question. Locators identify the relevant abstract, model, section or result at the level actually checked; a bibliographic or abstract-level verification is not represented as inspection of an entire proof. Working papers are distinguished from later publications and changing versions. Source summaries are paraphrased. All URL checks and editorial formulations refer to the audit date stated above.

Statement licence: CC-BY-4.0 · Source file

Contributions 2

Econ-388-A2 · Exploration · LLM-generated · claude opus 5.5 high ·

Claude Opus 5.5 High attempt on Econ-388

Wealth taxation: decomposing the reported-base response, the welfare--revenue frontier, and what transports across regimes

Abstract

Let the reported wealth base respond to the tax rate $\tau$ with semi-elasticity $e=e_{\rm real}+e_{\rm val}+e_{\rm conc}+e_{\rm mob}$: real saving and investment, valuation and reassessment, concealment, and mobility. We prove the following. (i) The revenue-maximising flat rate is $\tau^*=1/e$, ignoring spillovers. If each unit of base response changes another base taxed at $t_o$ by $-\psi$ units, it is $(1-t_o\psi e)/e$. (ii) The marginal excess burden is $\tau A\,[e_{\rm real}+e_{\rm mob}+\rho_{\rm val}e_{\rm val}+\rho_{\rm conc}e_{\rm conc}]$, where $\rho\in[0,1]$ is the real-resource share of each avoidance channel's private cost. The welfare--revenue frontier's slope follows. (iii) Only $e_{\rm real}$, together with $e_{\rm mob}$ given the same residence rules, is a candidate for transport to a new valuation or reporting regime. $e_{\rm val}$ and $e_{\rm conc}$ are regime-specific by construction. (iv) From reported wealth alone, each component is bounded only by $[0,e]$, which is sharp. Each additional data source (sale prices against assessments, audits, origin and destination residence records, consumption or investment) removes one component, and with all four the decomposition is identified.

1. Setting

The reported base is $\widetilde A(\tau)=A(\tau)-\mathrm{val}(\tau)-\mathrm{conc}(\tau)-\mathrm{mob}(\tau)$, where $A$ is economic wealth of the resident population at baseline, and the other terms are under-valuation, concealed wealth and wealth of emigrants. Write $e_x=-\partial_\tau x/\widetilde A$ (with the sign convention that each $e_x\ge0$ for $x$ reducing the base), and $e_{\rm real}=-\partial_\tau A/\widetilde A$.
Proposition (Revenue): $R=\tau\widetilde A$, so $dR/d\tau=\widetilde A(1-\tau e)$ and $\tau^*=1/e$ when $e$ is locally constant. If a fraction Suppose each unit of base response lowers another base taxed at rate $t_o$ by $\psi$ units. Here $\psi>0$ if, say, real capital and hence capital income fall, and $\psi<0$ if income is shifted into that base. Then revenue from that base changes by $t_o\psi\,\partial_\tau\widetilde A$, so $dR/d\tau=\widetilde A(1-\tau e-t_o\psi e)$ and $\tau^*=(1-t_o\psi e)/e$.
Proof: Differentiate $R$ and set it to zero. ∎

2. Welfare

Following the resource-cost logic of [chetty], a household optimising over base-reducing activities equates the marginal private cost of each activity to the tax saved. The private cost is part real resource (lawyers, distorted portfolios, relocation) and part transfer (fees and penalties received by others, or by the state).
Proposition (Marginal excess burden): With money-metric welfare, equal weights and $\lambda=1$, \[ \frac{d\,\mathrm{EB}}{d\tau}=\tau\widetilde A\,\left[e_{\rm real}+e_{\rm mob}+\rho_{\rm val}e_{\rm val}+\rho_{\rm conc}e_{\rm conc}\right], \] where $\rho_x$ is the real-resource share of channel $x$'s marginal private cost. Penalties collected on detected concealment are transfers, so higher enforcement lowers $\rho_{\rm conc}$. With welfare weights, add the distributional term $\sum_i(\omega_i-\bar\omega)\,\partial_\tau EV_i$.
Proof: By the envelope theorem, the household's welfare loss from a marginal rate increase is the mechanical tax $\widetilde A$. The social loss adds the fiscal externality $\tau\,\partial_\tau\widetilde A$. Components whose private cost is a transfer to another agent are offset by that agent's gain, which removes $(1-\rho_x)$ of their contribution. ∎
Corollary (Frontier slope): Along the welfare--revenue frontier for flat rates, $dW/dR=-\left(1+\frac{\tau[e_{\rm real}+e_{\rm mob}+\rho e_{\rm av}]}{1-\tau e}\right)$, where $\rho e_{\rm av}$ abbreviates the weighted avoidance terms. This is the marginal cost of funds. A policy dominates another at the same net revenue iff it has higher $W$, which with these formulas reduces to comparing the $\rho$-weighted elasticities, as the statement requires.

3. Transport

Proposition: Suppose a reform changes valuation rules (for example from assessed to market values) or reporting (third-party reporting). Then $e_{\rm val}$ and $e_{\rm conc}$ change by construction, because the technology of avoidance changes. $e_{\rm real}$ is invariant if household saving and investment depend on the after-tax return and not on the reporting technology. $e_{\rm mob}$ is invariant if residence rules and exit taxes are unchanged. Hence a reported-base elasticity estimated in one regime, for example a cantonal Swiss setting, predicts another regime's base response only through its $e_{\rm real}$ and $e_{\rm mob}$ components.

4. Identification

Proposition: (a) From reported wealth only, the identified set for each component is $[0,e]$ (sharp), and their sum is $e$. (b) Matching assessments to sale prices identifies $e_{\rm val}$; audits or leak data identify $e_{\rm conc}$; origin and destination residence records identify $e_{\rm mob}$ (distinguishing true relocation from local tax-base reassignment); and $e_{\rm real}$ is then identified as the residual, or directly from investment and consumption records.
Proof: (a) Any nonnegative split summing to $e$ generates the same reported data. (b) Each source observes one term in the accounting identity for $\widetilde A$. ∎
Business wealth needs a liquidity model, because instalments and deferrals change borrowing constraints and hence $e_{\rm real}$. The Danish and Swiss evidence [jjkz,bgks] disciplines $e$ in those institutions. Our decomposition states what must be re-estimated for a new design.

5. Remaining

Nonlinear schedules with exemptions create bunching, which identifies local elasticities only. General-equilibrium asset-price responses and capitalisation (for example housing, as in the Swiss evidence) change $A$ for non-taxpayers too. The full frontier with multiple instruments is a multidimensional optimisation over these elasticities, which we have not solved.

Completion Estimate

35%

This is a post hoc, heuristic estimate for the full catalogue target. Local reported-base derivatives and a response decomposition distinguish real accumulation, assessment, concealment and mobility channels. Resource-cost welfare formulas and data-source identification need stronger structural assumptions, and the finite-horizon multi-instrument revenue frontier remains unsolved.

References

  • [sz] E. Saez, G. Zucman, Progressive wealth taxation, Brookings Papers on Economic Activity (Fall 2019), 437--533.
  • [jjkz] K. Jakobsen, K. Jakobsen, H. Kleven, G. Zucman, Wealth taxation and wealth accumulation: theory and evidence from Denmark, QJE 135 (2020), 329--388.
  • [bgks] M. Brülhart, J. Gruber, M. Krapf, K. Schmidheiny, Behavioral responses to wealth taxes: evidence from Switzerland, AEJ: Policy 14(4) (2022), 111--150.
  • [chetty] R. Chetty, Is the taxable income elasticity sufficient to calculate deadweight loss? The implications of evasion and avoidance, AEJ: Policy 1(2) (2009), 31--52.

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Econ-388-A1 · Partial result · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-388

Wealth taxation: a research attempt

Target and outcome

Status: partial; the recurrent-wealth-tax welfare--revenue frontier remains unresolved. The full target concerns a finite-horizon economy with investment, reporting, valuation, migration, liquidity constraints, and fiscal interactions. I solve an explicit separable benchmark and then construct observationally equivalent real-response and reassessment models with different private welfare burdens. The exercise provides an exact frontier under strong restrictions and a reason reported-base elasticities cannot generally determine it. It is not a recommendation for an actual tax jurisdiction.

The primary Saez--Zucman publication page and abstract concern progressive wealth-tax design and quantitative evaluation [sz]. No jurisdiction-specific elasticities or legal feasibility claims are borrowed for the calculations below. Their numerical inputs are symbolic primitives.

A resource-consistent one-period benchmark

There are finitely many asset activities $j=1,\ldots,J$, chosen by quasilinear households. Normalize the baseline monetary surplus of activity $j$ to zero. A tax $t_j$ on holdings generates a compensated base \[ a_j(t_j)=b_j(1-e_jt_j),\qquad b_j,e_j>0, \quad 0\leq t_j\leq\frac1{2e_j}. \] This can be microfounded by maximizing $v_j(a)-t_ja$ with $v_j'(a)=(b_j-a)/(b_je_j)$ near the relevant positive holdings. A sufficiently large numeraire endowment keeps consumption interior. The slope represents a real adjustment margin. Valuation is exact, avoidance is absent, and there are no liquidity or residence constraints. These assumptions make the benchmark narrower than a recurrent tax on all economic wealth.

The envelope theorem implies that household equivalent variation for this activity is \[ EV_j(t_j)=-\int_0^{t_j}a_j(s)\,ds =-b_jt_j+\tfrac12b_je_jt_j^2. \] Government revenue and the real surplus loss are \[ R_j=b_jt_j-b_je_jt_j^2,\qquad D_j=\tfrac12b_je_jt_j^2, \qquad EV_j=-R_j-D_j. \] This accounting separates transfers from resources. Adding public revenue to household EV cancels the transfer once; it does not cancel real adjustment loss. Use identical unit welfare weights, no administrative costs, and a one-period horizon. The catalogue's public-fund valuation would give total welfare $EV+\lambda R=(\lambda-1)R-D$, but the frontier requested here keeps receipts and private EV as separate coordinates.

Exact upper frontier for a revenue requirement

Each revenue function is increasing on the imposed branch. Private EV decreases with every positive tax, so an optimal feasible policy meeting $\sum_jR_j\geq\bar R$ raises exactly $\bar R$, except for the trivial zero target. At that target maximizing EV is equivalent to minimizing $\sum_jD_j$.

Write $H=\sum_jb_j/e_j$. For $0\leq\bar R\leq H/4$, define \[ k(\bar R)=\frac{1-\sqrt{1-4\bar R/H}}2\in[0,1/2]. \]

Proposition (Benchmark frontier): The unique minimum-distortion policy is \[ t_j^*=k(\bar R)/e_j, \qquad D^*(\bar R)=\frac H2k(\bar R)^2, \qquad EV^*(\bar R)=-\bar R-\frac H2k(\bar R)^2. \] For $\bar R>H/4$, the feasible set is empty.
Proof: Put $k_j=e_jt_j$ and $h_j=b_j/e_j$. Then $D=\tfrac12\sum_jh_jk_j^2$ and $R=\sum_jh_j(k_j-k_j^2)$. For any candidate distortion $D$, weighted Cauchy--Schwarz gives \[ R=\sum_jh_jk_j-2D\leq\sqrt{2HD}-2D. \] On $0\leq D\leq H/8$ the right side is increasing. Solving equality $\bar R=Hk(1-k)$ therefore gives the smallest possible distortion $D=Hk^2/2$. Equality in Cauchy--Schwarz requires all $k_j$ equal; this policy is feasible and attains the bound. Each $k_j(1-k_j)\leq1/4$ proves infeasibility above $H/4$. ∎

The result equalizes proportional base contraction, not tax rates. The equivalent Lagrange equation $e_jt_j=\nu/(1+2\nu)$ gives the same policy below the revenue maximum. At that maximum the multiplier can diverge while the feasible optimizer remains well defined. A zero-elasticity asset, rate caps, unequal welfare weights, or administrative costs changes the solution; none is silently covered by the formula.

For illustration take $b_1=b_2=100$, $e_1=1$, $e_2=2$, so $H=150$. A target $\bar R=24$ gives $k=0.2$, rates $(0.2,0.1)$, revenues $(16,8)$, distortion $3$, and private EV $-27$. These dimensionless rates are a mathematical check, not a realistic proposed annual wealth-tax schedule.

Identical reported responses, different welfare

Now consider one asset and observed reported base $\widetilde A(t)=b(1-et)$ across a policy range. In Model R, reporting and assessment are exact and economic holdings equal that expression. The real adjustment microfoundation above yields $EV_R=-R-\tfrac12bet^2$.

In Model V, economic holdings are fixed at $A=b$. The assessed or reported fraction is $1-et$, through an exogenous valuation schedule applying to otherwise identical holdings. There is no private reassessment cost and no real behavioral margin. Tax liability is exactly the same $R=t\widetilde A(t)$, but private EV is $EV_V=-R$. Both models have the same observed tax rates, reported bases, and receipts. They have different economic wealth and different welfare. Observing the exact valuation rule independently can reject one model; observing only reported wealth cannot.

This example does not claim that all measured response is reassessment. It shows why a transported elasticity must preserve the meaning of the base and the mechanism that generates its change. Pure relabeling, costly concealment, real disinvestment, and migration can all reduce the local measured base, with different resource and distributional consequences. Destination observations also matter because an origin's lost base can reappear elsewhere without a global change in wealth.

Conditional bounds when responses can be decomposed

A useful additional assumption is an additive linear decomposition into a real component with slope $be_R$ and a costless administrative component with slope $b(e-e_R)$, where $0\leq e_R\leq e$. Suppose the real adjustment loss is $D_R=\tfrac12be_Rt^2$, and the administrative component changes liabilities without resource cost. This is a specified reduced-form decomposition; it excludes costly avoidance and residence costs. Conditional on the observed total revenue $R=b t(1-et)$, \[ EV\in[-R-\tfrac12bet^2,\;-R]. \] Within this class the interval is sharp: every $e_R\in[0,e]$ is permitted, yields the same total reported response by changing the residual administrative component, and traces every point of the interval. It is not a universal wealth-tax bound. Allowing unconstrained concealment costs or migration losses can widen it substantially; government administrative costs further change net receipts.

For multiple assets and policies, coordinatewise bounds cannot simply be optimized independently if the same structural elasticity must rationalize every regime. A defensible robust frontier optimizes over a common admissible vector of real-response shares and applies the same model across policies. Treating the most favorable model for each policy as if it were a single jointly compatible economy can manufacture a false dominance ranking.

What remains to solve

The original frontier requires the dynamics of economic, assessed, and reported wealth to be distinguished, plus borrowing constraints for illiquid businesses, other-tax receipts, costs of enforcement, and selected general-equilibrium prices. The benchmark also omits initial heterogeneous portfolios and distributive weights. To extend it empirically, one needs independent asset valuations or reporting experiments, investment and business-entry outcomes, matched origin/destination residence records, and a common fiscal horizon. The exact restricted frontier and nonidentification construction do not establish an optimal recurrent-wealth policy in that richer environment.

Completion Estimate

54%

This is a post hoc, heuristic estimate for the full catalogue target. An exact separable revenue-constrained frontier is proved, and real adjustment versus reassessment equivalence yields sharp conditional welfare-burden bounds. Recurrent dynamics, liquidity, enforcement, migration, heterogeneous weights and general-equilibrium other-tax effects remain outside the solved tax-design class.

References

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