## Abstract

This paper examines the effect on the elasticity of taxable income for individuals in couples, where there is no income splitting for tax purposes but joint decisions are taken regarding taxable incomes. Two approaches are considered. First, the effects of minimising the total tax increase arising from a marginal rate increase are examined. Second, the paper considers the effects of joint utility maximisation. In both cases, analytical results suggest the possibility that empirical taxable income elasticity estimates obtained while ignoring the distinction between individuals and couples can be biased, sometimes substantially. Thus, in the presence of individual income taxation, both taxable income elasticity and revenue response estimates could be expected to be quite different when those estimates are obtained while erroneously treating the behaviour of members of couple households as if they were separate individuals.

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## Notes

- 1.
The properties and literature are reviewed by Saez et al. (2012).

- 2.
On the approach to optimal taxation using the ETI, see Saez (2001) and Creedy (2015). The modification needed in the case of joint taxation with income splitting for couples is examined by Bach et al. (2012). The optimal taxation of couples is also examined by Kleven et al. (2009), using the elasticity of taxable income concept.

- 3.
- 4.
For extensive discussion of this function and its revenue flexibility properties, see Creedy and Gemmell (2006).

- 5.
This uses \(\frac{dT_{1}}{d\tau _{1}}=z_{1}\left( 1+\eta _{z_{1},\tau _{1}}\right) \) and \(dT_{2}=\frac{T_{2}}{\tau _{1}}\eta _{z_{2},\tau _{1}}\). An equivalent expression for the response of \(z=z_{1}+z_{2}\) to \(\tau _{1}\) is given by: \(\eta _{z,\tau _{1}}=\left( \frac{z_{1}}{z}\right) \eta _{z_{1},\tau _{1}}+\left( \frac{z_{2}}{z}\right) \eta _{z_{2},\tau _{1}}\).

- 6.
Alternatively, the couple may be concerned with the reduction in the household’s net-of-tax income, \(\sum _{i=1}^{2}\left\{ y_{i}-T\left( y_{i}\right) \right\} \), by reducing or increasing gross income, \( \sum _{i=1}^{2}y_{i}\), or reducing \(\sum _{i=1}^{2}T\left( y_{i}\right) \), or reallocating income across household members. This potentially includes responses that generate an increase in total net-of-tax income if income effects on gross income are sufficiently large. These cases are not examined here.

- 7.
Additionally, as Saez et al. (2012) point out, avoidance and evasion responses that are

*not*associated with income transfers to other household members, may also have limited welfare effects if they simply represent transfers among taxpayers in general. - 8.
Clearly, if \(\beta =0\), Eq.(10) reduces to \(\eta _{T,\tau _{1}}=\frac{T_{1}}{T}\left\{ 1+\eta _{z_{1},\tau _{1}}\right\} +\frac{T_{2}}{ T}\eta _{z_{2},\tau _{1}}^{^{\prime }}\), capturing the two separate revenue response components for each member.

- 9.
The only study to allow for income effects for individual members of couple households, Gelber (2014) for Sweden, also found income effects to be small, around \(-0.05\) to \(-0.07\). These may be compared with compensated elasticities of taxable income estimates of around \(-0.41\) to \(-0.47\) (with uncompensated values of around \(-0.21~\) to \(-0.23\)).

- 10.
- 11.
In a recent extension of such collective models, Meier and Rainer (2015) show how ‘optimal’ (Pigouvian) tax rules for both cooperative and non-cooperative approaches to utility maximisation by couples can be derived. However, in common with the collective literature, tax responses are again restricted to labour supply choices.

- 12.
If, alternatively, the two components of

*U*in (18) are additive, income sharing has no effect on the solutions for \(z_{1}\) and \( z_{2}\). - 13.
In his Swedish study, Gelber (2014, p. 295) examines different responses by members of married couple households (hence obtaining separate estimates of \( \eta _{y_{1},1-\tau _1}\), \(\eta _{y_{2},1-\tau _{2}}\), \(\eta _{y_{1},1-\tau _{2}}\) and \(\eta _{y_{2},1-\tau _{1}}\)), but excludes non-married and single-person households. Also, in the context of estimating structural female labour supply models for the UK, Blundell et al. (2016; section 7.3) find that labour supply elasticities for women in couples (with and without children) can differ substantially from equivalent elasticity estimates for single women (with and without children). Other studies to focus specifically on labour supply responses to taxation for married couples and/or parents include Eissa (1995) and LaLumia (2008) for the USA, and Stephens and Ward-Batts (2004) for the UK.

- 14.
Write \(\xi =\frac{k_{1}}{k_{2}}=\frac{1+\frac{1}{\varepsilon _{2}}}{ \varepsilon _{2}}\). The coefficient, \(\varepsilon _{2}\), can be obtained as the appropriate root of the quadratic \(\xi \varepsilon _{2}^{2}-\varepsilon _{2}-1=0\). Using the expression for \(k_{1}\), and the solution for \( \varepsilon _{2}\), the coefficient, \(\varepsilon _{1}\), can also be obtained as the root of another quadratic function, given by \(\left( 1+\frac{ k_{1}\varepsilon _{2}}{1+\frac{1}{\varepsilon _{2}}}\right) \varepsilon _{1}^{2}+\left( 1-k_{1}\right) \varepsilon _{1}-k_{1}=0\). The separate parameters, \(\varepsilon _{1}\) and \(\varepsilon _{2}\), can therefore be recovered from estimation of (31).

- 15.
In terms of impacts on tax revenue

*levels*, this can translate into a substantive difference. For example, with \(y_{1}=\$120,000\) and \( y_{2}=\$65,000\), and other parameters as described above, person 1 pays tax of \(\$35,150\) when the top tax rate is 0.38. Raising this tax rate by 1%, to 0.3838, has a mechanical effect of raising tax revenue from person 1 by \(\$19{\dot{0}}\). Where person 1 is a single individual, the behavioural effect (of an ETI = 0.5 for person 1) is to cut this increase to \(\$50\), whereas if person 1 is coupled with person 2 (ETI = 0.25), the combined behavioural effect reduces the tax revenue increase to only \(\$39\), due to the reallocation of some taxable income to person 2 in this latter case. Clearly, this difference is non-trivial. - 16.
- 17.
For example, when \(\varepsilon _{1}=0.5\) and \(\varepsilon _{2}=0.25,\) then \( (\varepsilon _{1}\varepsilon _{2})^{2}=0.0156\), while if \(\varepsilon _{1}=0.3\) and \(\varepsilon _{2}=0.15,\)\((\varepsilon _{1}\varepsilon _{2})^{2}=0.0020\).

## References

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*European Economic Review*,*56*, 1055–1069.Blomquist, S., & Selin, H. (2010). Hourly wage rate and taxable labor income responsiveness to changes in marginal tax rates.

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## Acknowledgements

This paper is part of a larger project on ‘Improving New Zealand’s Tax Policy via International Tax Transfer Model Benchmarking’, funded by an Endeavour Research Grant from the Ministry of Business, Innovation and Employment (MBIE) and awarded to the Chair in Public Finance. We are grateful to two referees for comments on an earlier version.

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## Appendix: Relationships between revenue elasticities

### Appendix: Relationships between revenue elasticities

This Appendix demonstrates why the four profiles in Fig. 3 are approximately linear. To see why this is the case, it is useful to consider each profile separately. Consider first \(\eta _{T_{1},\tau _{1}}\). From Eq. (35), since \(\eta _{y_{1},1-\tau _{1}}=\varepsilon _{1}\) for a single individual, then

Hence, it can be seen that the slope of the relationship between \(\eta _{T_{1},\tau _{1}}\) and \(\varepsilon _{1}\) is constant with respect to changes in \(\varepsilon _{1}\), given by \(-\left( \frac{\tau _{1}}{1-\tau _{1} }\right) \left( \eta _{T_{1},y_{1}}\right) \) where, from (37), \(\eta _{T_{1},y_{1}}=\frac{y_{1}}{y_{1}-a_{k}^{*}}\). The single-person \(\eta _{T_{1},\tau _{1}}\) profile is therefore exactly linear, given a fixed tax structure and the individual’s income.

For two individuals in a couple, first consider person 1, where from (35) and (25):

For most plausible values of \(\varepsilon _{1}\) and \(\varepsilon _{2}\), it can be expected that \((\varepsilon _{1}\varepsilon _{2})^{2}\) is small.^{Footnote 17} Thus, from Eq. (A.2), if \((\varepsilon _{1}\varepsilon _{2})^{2}\approx 0\), the variation between \(\eta _{T_{1},\tau _{1}}\) and \(\varepsilon _{1}\) reduces to:

Following a similar process for person 2, from (41 ) and (25) it can be shown that:

and:

if \((\varepsilon _{1}\varepsilon _{2})^{2}\approx 0\). However, unlike the case for \(\eta _{T_{1},\tau _{1}}\) in (A.3), the approximation for \( \eta _{T_{2},\tau _{1}}\) does not display a simple linear relationship with \( \varepsilon _{1}\). That is, both the exact specification in (A.4) and the approximation in (A.5) suggest nonlinear relationships between \( \eta _{T_{2},\tau _{1}}\) and \(\varepsilon _{1}\).

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Creedy, J., Gemmell, N. The elasticity of taxable income of individuals in couples.
*Int Tax Public Finance* **27, **931–950 (2020). https://doi.org/10.1007/s10797-019-09581-6

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### Keywords

- Income taxation
- Taxable income
- Elasticity of taxable income

### JEL Classification

- H24
- H31
- H26