The elasticity of taxable income of individuals in couples

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.

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

$$\begin{aligned} \eta _{T_{1},\tau _{1}}=\eta _{T_{1},\tau _{1}}^{\prime }-\left( \frac{\tau _{1}}{1-\tau _{1}}\right) \left( \eta _{T_{1},y_{1}}\right) \varepsilon _{1} \end{aligned}$$

(A.1)

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):

$$\begin{aligned} \eta _{T_{1},\tau _{1}}=\eta _{T_{1},\tau _{1}}^{\prime }-\left( \frac{\tau _{1}}{1-\tau _{1}}\right) \left( \eta _{T_{1},y_{1}}\right) \varepsilon _{1} \left[ \frac{(1+\varepsilon _{1})(1+\varepsilon _{2})}{(1+\varepsilon _{1})(1+\varepsilon _{2})-(\varepsilon _{1}\varepsilon _{2})^{2}}\right] \end{aligned}$$

(A.2)

For most plausible values of $\varepsilon _{1}$ and $\varepsilon _{2}$, it can be expected that $(\varepsilon _{1}\varepsilon _{2})^{2}$ is small.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:

$$\begin{aligned} \eta _{T_{1},\tau _{1}}\approx \eta _{T_{1},\tau _{1}}^{\prime }-\left( \frac{\tau _{1}}{1-\tau _{1}}\right) \left( \eta _{T_{1},y_{1}}\right) \varepsilon _{1} \end{aligned}$$

(A.3)

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

$$\begin{aligned} \eta _{T_{2},\tau _{1}}=\left( \frac{\tau _{1}}{1-\tau _{1}}\right) \left( \frac{y_{2}}{y_{2}-a_{s}^{*}}\right) \left( \frac{\varepsilon _{1}\varepsilon _{2}}{1+\varepsilon _{1}}\right) \left[ \frac{(1+\varepsilon _{1})(1+\varepsilon _{2})}{(1+\varepsilon _{1})(1+\varepsilon _{2})-(\varepsilon _{1}\varepsilon _{2})^{2}}\right] \quad \end{aligned}$$

(A.4)

and:

$$\begin{aligned} \eta _{T_{2},\tau _{1}}\approx \left( \frac{\tau _{1}}{1-\tau _{1}}\right) \left( \frac{y_{2}}{y_{2}-a_{s}^{*}}\right) \left( \frac{\varepsilon _{1}\varepsilon _{2}}{1+\varepsilon _{1}}\right) \end{aligned}$$

(A.5)

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 $\varepsilon _{1}$.