## Abstract

We study the optimal income taxation of couples. We determine the resulting intra-family labor supply allocation and its implication for the choice of the tax unit (individual versus joint taxation). We provide a general condition for full joint taxation to arise. We also study how the spouses’ respective labor supply decisions are distorted when the condition does not hold. In particular, we show that, depending on the pattern of mating, the celebrated result according to which the spouse with the more elastic labor supply faces the lower marginal tax rates may or may not hold in our setting.

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## Notes

The issue of the choice of the tax base is only briefly addressed in his Proposition 7.

This model of the couple is a fairly general reduced form of the unitary model. It is consistent with the existence of “private” and “public” goods in the household and with the determination of consumption and labor supply levels through a bargaining process (where the weights are exogenous and not affected by the tax policy). From that perspective, one can think of

*U*(*x*,ℓ_{ w },ℓ_{ y }) as the value function associated with the problem of allocating a household budget*x*to the various consumption goods, given labor supply levels ℓ_{ w }and ℓ_{ y }. In other words,*U*(*x*,ℓ_{ w },ℓ_{ y }) specifies the*maximum*level of the households objective (*e.g.*, a weighted sum of utilities or a Nash product) given labor supplies and given its total (after tax budget). Using this reduced form rather than the original household objective in the optimal tax problem does not involve any loss of generality as long as differential commodity taxation is not possible, for instance because individual consumption levels (and thus the allocation of goods within the couple) are not publicly observable.This is admittedly a highly stylized and incomplete typology. In particular, it abstracts from the tax treatment of children and relies on the assumption that there are no single individuals.

Which does not necessarily imply that

*all*implementing tax functions are unitary.When marginal taxes are different from zero, leisure-labor and/or domestic labor-market labor tradeoffs are distorted. However, the intra-family allocation of labor supply (as specified by problem (2)) remains undistorted when marginal tax rates are the same for both spouses.

Similarly, there will be a distortion towards less

*y*_{ h }in the \(\left( y_{w},y_{h}\right) \) tradeoff when:$$ \left. \frac{{\rm d}T ( y_{h},y_{w})}{{\rm d}y_{w}}\vert_{y_{w} + y_{h}=GI}=\frac{ \partial T( y_{h},y_{w}) }{\partial y_{w}}-\frac{\partial T(y_{h},y_{w})}{\partial y_{h}}<0.\right. $$This statement is related to Proposition 5 of Brito et al. (1990). In our set up, their proposition implies that if \(\partial T\left( y_{w}^{i},y_{h}^{i}\right) /\partial y_{w}^{i}>\) \(\partial T\left( y_{w}^{i},y_{h}^{i}\right) /\partial y_{h}^{i}\) (respectively, < ), then there exists a non empty set of couples

*K*⊂*N*such that for*k*∈*K*,*λ*^{ki}> 0 and \({\rm MRS}_{y_{w},y_{h}}^{i}<{\rm MRS}_{y_{w},y_{h}}^{k}\) (respectively, >).The ratio between spouses productivities is typically referred to a “gap” in the empirical literature. We follow this tradition here.

Formally, the Frisch labor supply elasticity

*ε*is here defined by:$$ \varepsilon =\frac{U_{\ell }}{\ell \left( U_{\ell \ell }-\frac{\left( U_{cl}\right) ^{2}}{U_{cc}}\right) }, $$where subscripts denote partial derivatives.

With our assumption couples can be ordered by the >

_{ F }relation defined by Brett (2007), and the determination of the pattern of binding incentive constraints is dramatically simplified.This assumption is stronger than necessary, but it dramatically simplifies notation. All our qualitative results go through if we assume simply that only downward incentive constraints are binding. To see this, observe that the pairwise comparisons of MRS we perform are valid also when

*j*≠*i*+ 1. The translation into marginal tax rates is slightly more complicated because we may have that more than one IC constraint towards a given type is binding. However, we may recall from Eq. 11 that the total effect is obtained by adding the pairwise effects. Consequently, when all these effects go in the same direction, the study of the pairwise effects is sufficient.

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## Acknowledgements

This paper has been presented at the CESifo Public Sector Economics Conference, the IIPF Congress (Warwick) and the HIM/Max Planck Institute workshop “Incentives, Efficiency, and Redistribution in Public Economics”. We thank the participants and particularly, Bas Jacobs and Johannes Spinnewijn, for their comments. We are also grateful to the two referees and the editor, Alessandro Cigno, for their detailed and constructive remarks. Last but not least, we thank Hélène Couprie for providing us with data and estimations regarding the wage gap between spouses in French couples.

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*Responsible editor:* Alessandro Cigno

## Appendices

### Appendix

### A Proof of Proposition 1

Combining Eqs. 4 and 10 yields

so that

Numerator and denominator of Eq. 15 are negative. Consequently this property is equivalent to

Simplifying and rearranging the last inequality establishes Proposition 1.

### B Proof of Proposition 2

When preferences are represented by (12) one has:

so that

Rearranging the terms and making use of the definition of *P*
^{i} establishes the proposition.

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Cremer, H., Lozachmeur, JM. & Pestieau, P. Income taxation of couples and the tax unit choice.
*J Popul Econ* **25**, 763–778 (2012). https://doi.org/10.1007/s00148-011-0376-6

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DOI: https://doi.org/10.1007/s00148-011-0376-6

### Keywords

- Optimal income taxation
- Tax unit
- Household labor supply

### JEL Classification

- H21
- H31
- D10