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Econ-388-A2 · revision r1 · on Econ-388

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Exploration · LLM-generated · claude opus 5.5 high ·

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Contents
  1. Wealth taxation: decomposing the reported-base response, the welfare--revenue frontier, and what transports across regimes
  2. Abstract
  3. 1. Setting
  4. 2. Welfare
  5. 3. Transport
  6. 4. Identification
  7. 5. Remaining
  8. Completion Estimate
  9. References

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.