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Reducing the dimensionality of a selfishly optimal nonlinear income tax problem

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Abstract

Röell (Voting over nonlinear income tax schedules, unpublished manuscript, 2012) considers a political economy version of the finite-type Mirrlees (Rev Econ Stud 38:175–208, 1971) nonlinear income tax problem in which each type proposes its selfishly optimal tax schedule with majority rule used to select among them. We show that it is possible to solve a selfishly optimal nonlinear income tax problem by first solving for the optimal consumptions using a reduced-form problem that only involves these variables and then using the optimal consumptions in two recursion formulae to compute the optimal incomes. The analysis extends the methodology introduced in Weymark (J Publ Econ 30:199–217, 1986b) so as to handle problems in which not all of the adjacent incentive constraints bind in the same direction.

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Notes

  1. Chambers (1989) generalizes Weymark’s analysis to the case in which preferences are additively separable but at the cost of having a recursion formula in which not all of the variables are defined explicitly.

  2. For simplicity, Weymark (1986b) does not take account of the nonnegativity constraints on incomes. Here, we explicitly consider them.

  3. In Weymark (1987), the solution to his reduced-form problem is used to determine the comparative static properties of the optimal allocation with respect to the exogenous parameters.

  4. We are implicitly assuming that taxation is only used for redistributive purposes. The qualitative features of our analysis are unchanged if the government needs to use a fixed amount R of the private good to provide public services. In this case, R would be added to the left-hand side of (7) and to the right-hand side of (9).

  5. Because the number of types is finite, there is not a unique tax schedule that generates an allocation that satisfies the self-selection constraints. This indeterminacy is of no consequence because it is the allocation, not the tax schedule that generates it, that is of interest.

  6. If \(y^k_1 > 0\), at this point we could obtain the desired contradiction by marginally decreasing everybody’s incomes by the same amount. The following argument applies whether or not \(y^k_1 > 0\).

  7. If \(y^k_1 > 0\), as with Lemma 2, a contradiction can be obtained by decreasing everybody’s incomes by a common small amount. This is not possible if \(y^k_1= 0\).

  8. With a continuum of types, there is necessarily bunching of types in an interval containing the proposer’s type (see Brett and Weymark 2017b). This need not be the case with a finite number of types.

  9. The parameter p can be interpreted as being the price of the consumption good.

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Acknowledgements

We are grateful for the comments of an anonymous referee. We are particularly grateful for a comment that allowed us to simplify some proofs.

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Correspondence to John A. Weymark.

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Brett, C., Weymark, J.A. Reducing the dimensionality of a selfishly optimal nonlinear income tax problem. Econ Theory Bull 6, 157–169 (2018). https://doi.org/10.1007/s40505-017-0131-6

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