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10-07-2021 | Original Paper

Optimal tax problems with multidimensional heterogeneity: a mechanism design approach

Authors: Laurence Jacquet, Etienne Lehmann

Published in: Social Choice and Welfare | Issue 1-2/2023

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Abstract

We propose a new method, that we call an allocation perturbation, to derive the optimal nonlinear income tax schedules with multidimensional individual characteristics on which taxes cannot be conditioned. It is well established that, when individuals differ in terms of preferences on top of their skills, optimal marginal tax rates can be negative. In contrast, we show that with heterogeneous behavioral responses and skills, one has optimal positive marginal tax rates, under utilitarian preferences and maximin.

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Appendix
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Footnotes
1
John Weymark has largely contributed to the literature on optimal income taxation, see e.g. Weymark (1986), Weymark (1987) and Brett and Weymark (2011).
 
2
Our paper studies the optimal tax system when individual characteristics, despite being observable by the tax authority, cannot be used as tags (Akerlof 1978), due to legal and/or horizontal equity reasons.
 
3
Our definition of “group” is identical to the one in Werning (2007), p. 13.
 
4
A smoothly increasing (decreasing) function is also called an increasing (decreasing) diffeomorphism for which the derivative maps the positive real line onto itself.
 
5
In (Jacquet and Lehmann 2021, Proposition 5), we show that the assumption of a smoothly-increasing-in-types allocation amounts to assuming: (i) twice differentiability of the tax function \(T(\cdot )\), that (ii) for all \((w,\theta )\in \mathbb {R}_+^*\times \Theta\), the second-order condition associated to the individual maximization program holds strictly and that (iii) for all \((w,\theta )\in \mathbb {R}_+^*\times \Theta\), the function \(y\mapsto \mathscr {U}\left( y-T(y),y;w,\theta \right)\) admits a unique global maximum over \(\mathbb {R}_+\).
 
6
For instance, we never found cases where the second-order incentive-compatibility constraints were violated in the large set of simulations we run on US data with taxpayers differing in terms of gender and labor supply elasticities, see Jacquet and Lehmann (2021).
 
7
More precisely, in the left-hand side of Eq. (14a), the term \(-\frac{v_{y}\left[ w,\theta \right] }{w\cdot v_{yw}\left[ w,\theta \right] }\) which is equal to the ratio of \(\varepsilon (w,\theta )\) and \(\alpha (w,\theta )\) [see Eq. (35) in the Appendix], is weighted by the conditional density times the skill, \(W(w,\theta )\ f(W(y,\theta )|\theta )\). And, in the right-hand side of (14a), which encapsulates the mechanical and income effects, the weights are the conditional skill densities.
 
8
In Hellwig (2007), under a utilitarian criterion, positive optimal tax rates are obtained with more general preferences.
 
9
If the utility function \(u(\cdot )\) in (1) were parameterized by type w and \(\theta\) while \(v(\cdot )\) were simply parameterized by w, individuals who earn the same income would have distinct social marginal welfare weights. This could drive negative marginal tax rates. Similarly, if both \(u(\cdot )\) and \(v(\cdot )\) were parameterized by w and \(\theta\), one would also expect negative marginal tax rates. Let us stress that our method could not be used in this framework since the pooling function (10) cannot depend simultaneously on Y and C.
 
10
Hence function \({\underline{W}}(\cdot ,\theta )\) coincides with the pooling function \(W(\cdot ,\theta )\).
 
11
Indeed, at \(m=0\), \(\mathscr {Y}^R_y\) does no longer depend on the direction R of the tax reform.
 
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Metadata
Title
Optimal tax problems with multidimensional heterogeneity: a mechanism design approach
Authors
Laurence Jacquet
Etienne Lehmann
Publication date
10-07-2021
Publisher
Springer Berlin Heidelberg
Published in
Social Choice and Welfare / Issue 1-2/2023
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI
https://doi.org/10.1007/s00355-021-01349-4

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