Proposition C.1 (Efficiency Loss with HSV Tax Function). Under joint taxation
of spousal incomes, described by (39), efficiency loss for
$\left(\upsilon_{m},\upsilon_{f}\right)$-couple from a small change in tax
progressivity $d\theta\approx 0$ is given by

$\frac{dD_{joint}\left(\upsilon_{m},\upsilon_{f}\right)}{d\theta}=\left[1-\lambda^{\frac{\sigma}{\sigma+\theta}}(1-\theta)^{\frac{\sigma}{\sigma+\theta}}\left(\upsilon_{m}+\upsilon_{f}\right)^{-\frac{\sigma\theta}{\sigma+\theta}}\right]\times\\\
\frac{\left[\lambda(1-\theta)^{1-\sigma-\theta}\left(\upsilon_{m}+\upsilon_{f}\right)^{\sigma}\right]^{\frac{1}{\sigma+\theta}}}{\sigma+\theta}\left[1+\frac{(1-\theta)\log\left(\lambda(1-\theta)\left(\upsilon_{m}+\upsilon_{f}\right)^{\sigma}\right)}{\sigma+\theta}\right]$

Under joint taxation of spousal incomes and linearized tax function,
efficiency loss for $\left(\upsilon_{m},\upsilon_{f}\right)$-couple from a
small reform $d\theta\approx 0$ is given by

$\frac{dD^{L}_{joint}\left(\upsilon_{m},\upsilon_{f}\right)}{d\theta}=\left[1-\lambda^{\frac{\sigma}{\sigma+\theta}}(1-\theta)^{\frac{\sigma}{\sigma+\theta}}\left(\upsilon_{m}+\upsilon_{f}\right)^{-\frac{\sigma\theta}{\sigma+\theta}}\right]\times\\\
\frac{\left[\lambda(1-\theta)^{1-\sigma-\theta}\left(\upsilon_{m}+\upsilon_{f}\right)^{\sigma}\right]^{\frac{1}{\sigma+\theta}}}{\sigma}\left[1+\frac{(1-\theta)\log\left(\lambda(1-\theta)\left(\upsilon_{m}+\upsilon_{f}\right)^{\sigma}\right)}{\sigma+\theta}\right]$

where $\lambda$ in both expressions above is given by

$\lambda=(1-g)^{\frac{\sigma+\theta}{\sigma}}(1-\theta)^{\frac{\theta}{\sigma}}\left[\frac{\int\left(\upsilon_{m}+\upsilon_{f}\right)^{\frac{\sigma}{\sigma+\theta}}d\varGamma\left(\upsilon_{m},\upsilon_{f}\right)}{\int\left(\upsilon_{m}+\upsilon_{f}\right)^{\frac{\sigma(1-\theta)}{\sigma+\theta}}d\varGamma\left(\upsilon_{m},\upsilon_{f}\right)}\right]^{\frac{\sigma+\theta}{\sigma}}$

Under separate taxation of spousal incomes, described by (40), efficiency loss
for $\left(\upsilon_{m},\upsilon_{f}\right)$-couple from a small change in tax
progressivity $d\theta\approx 0$ is given by

$\frac{dD_{sep}\left(\upsilon_{m},\upsilon_{f}\right)}{d\theta}=\\\
\left[1-\tilde{\lambda}^{\frac{\sigma}{\sigma+\theta}}(1-\theta)^{\frac{\sigma}{\sigma+\theta}}\upsilon_{m}^{-\frac{\sigma\theta}{\sigma+\theta}}\right]\frac{\left[\tilde{\lambda}(1-\theta)^{1-\sigma-\theta}\upsilon_{m}^{\sigma}\right]^{\frac{1}{\sigma+\theta}}}{\sigma+\theta}\left[1+\frac{(1-\theta)\log\left(\tilde{\lambda}(1-\theta)\upsilon_{m}^{\sigma}\right)}{\sigma+\theta}\right]+\\\
\left[1-\tilde{\lambda}^{\frac{\sigma}{\sigma+\theta}}(1-\theta)^{\frac{\sigma}{\sigma+\theta}}\upsilon_{f}^{-\frac{\sigma\theta}{\sigma+\theta}}\right]\frac{\left[\tilde{\lambda}(1-\theta)^{1-\sigma-\theta}\upsilon_{f}^{\sigma}\right]^{\frac{1}{\sigma+\theta}}}{\sigma+\theta}\left[1+\frac{(1-\theta)\log\left(\tilde{\lambda}(1-\theta)\upsilon_{f}^{\sigma}\right)}{\sigma+\theta}\right]$

Under separate taxation of spousal incomes and linearized tax function,
efficiency loss for $\left(\upsilon_{m},\upsilon_{f}\right)$-couple from a
small reform $d\theta\approx 0$ is given by

$\frac{dD^{L}_{sep}\left(\upsilon_{m},\upsilon_{f}\right)}{d\theta}=\\\
\left[1-\tilde{\lambda}^{\frac{\sigma}{\sigma+\theta}}(1-\theta)^{\frac{\sigma}{\sigma+\theta}}\upsilon_{m}^{-\frac{\sigma\theta}{\sigma+\theta}}\right]\frac{\left[\tilde{\lambda}(1-\theta)^{1-\sigma-\theta}\upsilon_{m}^{\sigma}\right]^{\frac{1}{\sigma+\theta}}}{\sigma}\left[1+\frac{(1-\theta)\log\left(\tilde{\lambda}(1-\theta)\upsilon_{m}^{\sigma}\right)}{\sigma+\theta}\right]+\\\
\left[1-\tilde{\lambda}^{\frac{\sigma}{\sigma+\theta}}(1-\theta)^{\frac{\sigma}{\sigma+\theta}}\upsilon_{f}^{-\frac{\sigma\theta}{\sigma+\theta}}\right]\frac{\left[\tilde{\lambda}(1-\theta)^{1-\sigma-\theta}\upsilon_{f}^{\sigma}\right]^{\frac{1}{\sigma+\theta}}}{\sigma}\left[1+\frac{(1-\theta)\log\left(\tilde{\lambda}(1-\theta)\upsilon_{f}^{\sigma}\right)}{\sigma+\theta}\right]$

where $\tilde{\lambda}$ in both expressions above is given by

$\tilde{\lambda}=(1-g)^{\frac{\sigma+\theta}{\sigma}}(1-\theta)^{\frac{\theta}{\sigma}}\left[\frac{\int\left(\upsilon_{m}^{\frac{\sigma}{\sigma+\theta}}+\upsilon_{f}^{\frac{\sigma}{\sigma+\theta}}\right)d\varGamma\left(\upsilon_{m},\upsilon_{f}\right)}{\int\left(\upsilon_{m}^{\frac{\sigma(1-\theta)}{\sigma+\theta}}+\upsilon_{f}^{\frac{\sigma(1-\theta)}{\sigma+\theta}}\right)d\varGamma\left(\upsilon_{m},\upsilon_{f}\right)}\right]^{\frac{\sigma+\theta}{\sigma}}$

Proof.

Joint Taxation of Spouses

The problem of couple characterized by preference parameters
$\left(\upsilon_{m},\upsilon_{f}\right)$ is given by

$\max_{c,y_{m},y_{f}}c-\frac{\upsilon_{m}}{\sigma+1}\left(\frac{y_{m}}{\upsilon_{m}}\right)^{\sigma+1}-\frac{\upsilon_{f}}{\sigma+1}\left(\frac{y_{f}}{\upsilon_{f}}\right)^{\sigma+1}$
(C.1) $\text{s.t.}~{}~{}~{}~{}~{}c=\lambda\left(y_{m}+y_{f}\right)^{1-\theta}$
(C.2)

Plugging (C.2) into (C.1) and maximizing with respect to $y_{m}$ and $y_{f}$,
I obtain the following first-order conditions:

$\left(\frac{y_{m}}{\upsilon_{m}}\right)^{\sigma}=\lambda(1-\theta)\left(y_{m}+y_{f}\right)^{-\theta}$
(C.3)
$\left(\frac{y_{f}}{\upsilon_{f}}\right)^{\sigma}=\lambda(1-\theta)\left(y_{m}+y_{f}\right)^{-\theta}$
(C.4)

Then, using $y_{m}/\upsilon_{m}=y_{f}/\upsilon_{f}$, derive the expressions
for the optimal taxable income:

$\tilde{y}_{m}=\lambda^{\frac{1}{\sigma+\theta}}(1-\theta)^{\frac{1}{\sigma+\theta}}\left(\upsilon_{m}+\upsilon_{f}\right)^{-\frac{\theta}{\sigma+\theta}}\upsilon_{m}$
(C.5)
$\tilde{y}_{f}=\lambda^{\frac{1}{\sigma+\theta}}(1-\theta)^{\frac{1}{\sigma+\theta}}\left(\upsilon_{m}+\upsilon_{f}\right)^{-\frac{\theta}{\sigma+\theta}}\upsilon_{f}$
(C.6)

Plugging these expressions into (C.2), I obtain the optimal consumption:

$\tilde{c}=\lambda^{\frac{\sigma+1}{\sigma+\theta}}(1-\theta)^{\frac{1-\theta}{\sigma+\theta}}\left(\upsilon_{m}+\upsilon_{f}\right)^{\frac{\sigma(1-\theta)}{\sigma+\theta}}$
(C.7)

Given the quasilinear preferences, the income effect on taxable income is
zero, and hence the Marshallian and Hicksian functions coincide. Next, using
(C.5) and (C.6), I obtain the compensated tax revenue function:

$T\left(\tilde{y}_{m},\tilde{y}_{f},\theta\right)=\tilde{y}_{m}+\tilde{y}_{f}-\lambda\left(\tilde{y}_{m}+\tilde{y}_{f}\right)^{1-\theta}=\\\
\lambda^{\frac{1}{\sigma+\theta}}(1-\theta)^{\frac{1}{\sigma+\theta}}\left(\upsilon_{m}+\upsilon_{f}\right)^{\frac{\sigma}{\sigma+\theta}}-\lambda^{\frac{\sigma+1}{\sigma+\theta}}(1-\theta)^{\frac{1-\theta}{\sigma+\theta}}\left(\upsilon_{m}+\upsilon_{f}\right)^{\frac{\sigma(1-\theta)}{\sigma+\theta}}$
(C.8)

Differentiating the compensated taxable income functions $\tilde{y}_{m}$ and
$\tilde{y}_{f}$ with respect to the tax progressivity parameter $\theta$, I
get

$\frac{d\tilde{y}_{m}}{d\theta}=-\upsilon_{m}\left[\frac{\lambda(1-\theta)}{\left(\upsilon_{m}+\upsilon_{f}\right)^{\theta}}\right]^{\frac{1}{\sigma+\theta}}\left[\frac{1}{(1-\theta)(\sigma+\theta)}+\frac{\log\left(\lambda(1-\theta)\left(\upsilon_{m}+\upsilon_{f}\right)^{\sigma}\right)}{(\sigma+\theta)^{2}}\right]$
(C.9)
$\frac{d\tilde{y}_{f}}{d\theta}=-\upsilon_{f}\left[\frac{\lambda(1-\theta)}{\left(\upsilon_{m}+\upsilon_{f}\right)^{\theta}}\right]^{\frac{1}{\sigma+\theta}}\left[\frac{1}{(1-\theta)(\sigma+\theta)}+\frac{\log\left(\lambda(1-\theta)\left(\upsilon_{m}+\upsilon_{f}\right)^{\sigma}\right)}{(\sigma+\theta)^{2}}\right]$
(C.10)

Next, differentiating the compensated tax revenue function (C.8) with respect
to taxable income, obtain

$\frac{\partial T}{\partial\tilde{y}_{m}}=\frac{\partial
T}{\partial\tilde{y}_{f}}=1-\lambda(1-\theta)\left(\tilde{y}_{m}+\tilde{y}_{f}\right)^{-\theta}=1-\lambda^{\frac{\sigma}{\sigma+\theta}}(1-\theta)^{\frac{\sigma}{\sigma+\theta}}\left(\upsilon_{m}+\upsilon_{f}\right)^{-\frac{\sigma\theta}{\sigma+\theta}}$
(C.11)

Using (C.9)-(C.10) and (C.11), I obtain the expression for marginal efficiency
loss for $\left(\upsilon_{m},\upsilon_{f}\right)$-couple under HSV tax and
transfer function:

$\frac{dD_{joint}\left(\upsilon_{m},\upsilon_{f}\right)}{d\theta}=-\left[\frac{\partial
T}{\partial\tilde{y}_{m}}\cdot\frac{d\tilde{y}_{m}}{d\theta}+\frac{\partial
T}{\partial\tilde{y}_{f}}\cdot\frac{d\tilde{y}_{f}}{d\theta}\right]=\left[1-\lambda^{\frac{\sigma}{\sigma+\theta}}(1-\theta)^{\frac{\sigma}{\sigma+\theta}}\left(\upsilon_{m}+\upsilon_{f}\right)^{-\frac{\sigma\theta}{\sigma+\theta}}\right]\times\\\
\frac{\left[\lambda(1-\theta)^{1-\sigma-\theta}\left(\upsilon_{m}+\upsilon_{f}\right)^{\sigma}\right]^{\frac{1}{\sigma+\theta}}}{\sigma+\theta}\left[1+\frac{(1-\theta)\log\left(\lambda(1-\theta)\left(\upsilon_{m}+\upsilon_{f}\right)^{\sigma}\right)}{\sigma+\theta}\right]$
(C.12)

Now I turn to the linearized program. Given the values of parameters
$\left(\theta,\lambda,\sigma,\upsilon_{m},\upsilon_{f}\right)$, denote the
solution to the couple’s problem by $\left(c^{*},y^{*}_{m},y^{*}_{f}\right)$.
Linearizing the budget constraint (C.2) around this point, I obtain

$c=\lambda(1-\theta)\left(y_{m}^{*}+y_{f}^{*}\right)^{-\theta}\left(y_{m}+y_{f}\right)+T^{*}$
(C.13)

where $T^{*}=\lambda\theta\left(y_{m}^{*}+y_{f}^{*}\right)^{1-\theta}+c^{*}$.

Next, I plug the linearized budget constraint (C.13) into the objective
function (C.1) and obtain the following first-order conditions:

$\left(\frac{y_{m}}{\upsilon_{m}}\right)^{\sigma}=\lambda(1-\theta)\left(y_{m}^{*}+y_{f}^{*}\right)^{-\theta}$
(C.14)
$\left(\frac{y_{f}}{\upsilon_{f}}\right)^{\sigma}=\lambda(1-\theta)\left(y_{m}^{*}+y_{f}^{*}\right)^{-\theta}$
(C.15)

Optimal taxable incomes in the problem with a linearized budget constraint are
given by

$\tilde{y}^{L}_{m}=\lambda^{\frac{1}{\sigma}}(1-\theta)^{\frac{1}{\sigma}}\left(y_{m}^{*}+y_{f}^{*}\right)^{-\frac{\theta}{\sigma}}\upsilon_{m}$
(C.16)
$\tilde{y}^{L}_{f}=\lambda^{\frac{1}{\sigma}}(1-\theta)^{\frac{1}{\sigma}}\left(y_{m}^{*}+y_{f}^{*}\right)^{-\frac{\theta}{\sigma}}\upsilon_{f}$
(C.17)

Differentiating the compensated taxable income functions $\tilde{y}^{L}_{m}$
and $\tilde{y}^{L}_{f}$ with respect to the tax progressivity parameter
$\theta$, I obtain

$\frac{d\tilde{y}^{L}_{m}}{d\theta}=-\left[\lambda(1-\theta)^{1-\sigma}\left(y_{m}^{*}+y_{f}^{*}\right)^{-\theta}\right]^{\frac{1}{\sigma}}\left[\frac{1}{\sigma}+\frac{(1-\theta)\log\left(y_{m}^{*}+y_{f}^{*}\right)}{\sigma}\right]\upsilon_{m}$
(C.18)
$\frac{d\tilde{y}^{L}_{f}}{d\theta}=-\left[\lambda(1-\theta)^{1-\sigma}\left(y_{m}^{*}+y_{f}^{*}\right)^{-\theta}\right]^{\frac{1}{\sigma}}\left[\frac{1}{\sigma}+\frac{(1-\theta)\log\left(y_{m}^{*}+y_{f}^{*}\right)}{\sigma}\right]\upsilon_{f}$
(C.19)

Plugging the optimal taxable income, (C.16) and (C.17), into these
expressions, I obtain

$\frac{d\tilde{y}^{L}_{m}}{d\theta}=-\upsilon_{m}\left[\frac{\lambda(1-\theta)^{1-\sigma-\theta}}{\left(\upsilon_{m}+\upsilon_{f}\right)^{\theta}}\right]^{\frac{1}{\sigma+\theta}}\left[\frac{1}{\sigma}+\frac{(1-\theta)\log\left(\lambda(1-\theta)\left(\upsilon_{m}+\upsilon_{f}\right)^{\sigma}\right)}{\sigma(\sigma+\theta)}\right]$
(C.20)
$\frac{d\tilde{y}^{L}_{f}}{d\theta}=-\upsilon_{f}\left[\frac{\lambda(1-\theta)^{1-\sigma-\theta}}{\left(\upsilon_{m}+\upsilon_{f}\right)^{\theta}}\right]^{\frac{1}{\sigma+\theta}}\left[\frac{1}{\sigma}+\frac{(1-\theta)\log\left(\lambda(1-\theta)\left(\upsilon_{m}+\upsilon_{f}\right)^{\sigma}\right)}{\sigma(\sigma+\theta)}\right]$
(C.21)

By construction, optimal taxable income in the problems with nonlinear and
linearized tax functions coincide, i.e. $\tilde{y}_{m}=\tilde{y}^{L}_{m}$ and
$\tilde{y}_{f}=\tilde{y}^{L}_{f}$. Therefore, $\partial
T/\partial\tilde{y}_{m}=\partial T/\partial\tilde{y}^{L}_{m}$ and $\partial
T/\partial\tilde{y}_{f}=\partial T/\partial\tilde{y}^{L}_{f}$. Using (C.11)
and (C.20)-(C.21), I obtain marginal deadweight loss for
$\left(\upsilon_{m},\upsilon_{f}\right)$-couple in the problem with a
linearized HSV tax function:

$\frac{dD^{L}_{joint}\left(\upsilon_{m},\upsilon_{f}\right)}{d\theta}=-\left[\frac{\partial
T}{\partial\tilde{y}^{L}_{m}}\cdot\frac{d\tilde{y}^{L}_{m}}{d\theta}+\frac{\partial
T}{\partial\tilde{y}^{L}_{f}}\cdot\frac{d\tilde{y}^{L}_{f}}{d\theta}\right]=\left[1-\lambda^{\frac{\sigma}{\sigma+\theta}}(1-\theta)^{\frac{\sigma}{\sigma+\theta}}\left(\upsilon_{m}+\upsilon_{f}\right)^{-\frac{\sigma\theta}{\sigma+\theta}}\right]\times\\\
\frac{\left[\lambda(1-\theta)^{1-\sigma-\theta}\left(\upsilon_{m}+\upsilon_{f}\right)^{\sigma}\right]^{\frac{1}{\sigma+\theta}}}{\sigma}\left[1+\frac{(1-\theta)\log\left(\lambda(1-\theta)\left(\upsilon_{m}+\upsilon_{f}\right)^{\sigma}\right)}{\sigma+\theta}\right]$
(C.22)

Finally, using the government budget constraint, I solve for $\lambda$ as a
function of policy parameters $(\theta,g)$ and primitives of the economy. The
government budget constraint under joint taxation of spouses is given by

$g\int\left(y_{m}+y_{f}\right)d\varGamma\left(\upsilon_{m},\upsilon_{f}\right)=\int\left(y_{m}+y_{f}\right)d\varGamma\left(\upsilon_{m},\upsilon_{f}\right)-\lambda\int\left(y_{m}+y_{f}\right)^{1-\theta}d\varGamma\left(\upsilon_{m},\upsilon_{f}\right)$
(C.23)

Solving for $\lambda$, I obtain

$\lambda=\frac{(1-g)\int\left(y_{m}+y_{f}\right)d\varGamma\left(\upsilon_{m},\upsilon_{f}\right)}{\int\left(y_{m}+y_{f}\right)^{1-\theta}d\varGamma\left(\upsilon_{m},\upsilon_{f}\right)}$
(C.24)

Finally, plugging (C.5) and (C.6), I derive the expression for the equilibrium
value of $\lambda$ under joint taxation of spouses:

$\lambda=(1-g)^{\frac{\sigma+\theta}{\sigma}}(1-\theta)^{\frac{\theta}{\sigma}}\left[\frac{\int\left(\upsilon_{m}+\upsilon_{f}\right)^{\frac{\sigma}{\sigma+\theta}}d\varGamma\left(\upsilon_{m},\upsilon_{f}\right)}{\int\left(\upsilon_{m}+\upsilon_{f}\right)^{\frac{\sigma(1-\theta)}{\sigma+\theta}}d\varGamma\left(\upsilon_{m},\upsilon_{f}\right)}\right]^{\frac{\sigma+\theta}{\sigma}}$
(C.25)

Note that, by construction, $\tilde{y}_{m}=\tilde{y}^{L}_{m}$ and
$\tilde{y}_{f}=\tilde{y}^{L}_{f}$, and hence the values of $\lambda$ in the
original and linearized programs coincide. This completes the derivation of
the expressions for reform-induced efficiency loss in the case of joint
taxation of spousal incomes.

Separate Taxation of Spouses

The problem of couple characterized by preference parameters
$\left(\upsilon_{m},\upsilon_{f}\right)$ is given by

$\max_{c,y_{m},y_{f}}c-\frac{\upsilon_{m}}{\sigma+1}\left(\frac{y_{m}}{\upsilon_{m}}\right)^{\sigma+1}-\frac{\upsilon_{f}}{\sigma+1}\left(\frac{y_{f}}{\upsilon_{f}}\right)^{\sigma+1}$
(C.26)
$\text{s.t.}~{}~{}~{}~{}~{}c=\tilde{\lambda}y_{m}^{1-\theta}+\tilde{\lambda}y_{f}^{1-\theta}$
(C.27)

Substituting (C.27) into (C.26) and maximizing with respect to $y_{m}$ and
$y_{f}$, I obtain the following first-order conditions:

$\left(\frac{y_{m}}{\upsilon_{m}}\right)^{\sigma}=\tilde{\lambda}(1-\theta)y_{m}^{-\theta}$
(C.28)
$\left(\frac{y_{f}}{\upsilon_{f}}\right)^{\sigma}=\tilde{\lambda}(1-\theta)y_{f}^{-\theta}$
(C.29)

Then, using
$y_{m}/y_{f}=\left(\upsilon_{m}/\upsilon_{f}\right)^{\frac{\sigma}{\sigma+\theta}}$,
I derive the expressions for the optimal taxable income:

$\tilde{y}_{m}=\tilde{\lambda}^{\frac{1}{\sigma+\theta}}(1-\theta)^{\frac{1}{\sigma+\theta}}\upsilon_{m}^{\frac{\sigma}{\sigma+\theta}}$
(C.30)
$\tilde{y}_{f}=\tilde{\lambda}^{\frac{1}{\sigma+\theta}}(1-\theta)^{\frac{1}{\sigma+\theta}}\upsilon_{f}^{\frac{\sigma}{\sigma+\theta}}$
(C.31)

Substituting these expressions into (C.27), obtain the optimal consumption:

$\tilde{c}=\tilde{\lambda}^{\frac{\sigma+1}{\sigma+\theta}}(1-\theta)^{\frac{1-\theta}{\sigma+\theta}}\left[\upsilon_{m}^{\frac{\sigma(1-\theta)}{\sigma+\theta}}+\upsilon_{f}^{\frac{\sigma(1-\theta)}{\sigma+\theta}}\right]$
(C.32)

Given the quasilinear preferences, the income effect on taxable incomes is
zero, and hence the Marshallian and Hicksian functions coincide. Next, using
(C.30) and (C.31), I obtain

$T\left(\tilde{y}_{m},\tilde{y}_{f},\theta\right)=\tilde{y}_{m}+\tilde{y}_{f}-\tilde{\lambda}\tilde{y}_{m}^{1-\theta}-\tilde{\lambda}\tilde{y}_{f}^{1-\theta}=\\\
\tilde{\lambda}^{\frac{1}{\sigma+\theta}}(1-\theta)^{\frac{1}{\sigma+\theta}}\left[\upsilon_{m}^{\frac{\sigma}{\sigma+\theta}}+\upsilon_{f}^{\frac{\sigma}{\sigma+\theta}}\right]-\tilde{\lambda}^{\frac{\sigma+1}{\sigma+\theta}}(1-\theta)^{\frac{1-\theta}{\sigma+\theta}}\left[\upsilon_{m}^{\frac{\sigma(1-\theta)}{\sigma+\theta}}+\upsilon_{f}^{\frac{\sigma(1-\theta)}{\sigma+\theta}}\right]$
(C.33)

Differentiating the compensated taxable income functions $\tilde{y}_{m}$ and
$\tilde{y}_{f}$ by parameter $\theta$, I obtain

$\frac{d\tilde{y}_{m}}{d\theta}=-\left[\tilde{\lambda}(1-\theta)\upsilon_{m}^{\sigma}\right]^{\frac{1}{\sigma+\theta}}\left[\frac{1}{(1-\theta)(\sigma+\theta)}+\frac{\log\left(\tilde{\lambda}(1-\theta)\upsilon_{m}^{\sigma}\right)}{(\sigma+\theta)^{2}}\right]$
(C.34)
$\frac{d\tilde{y}_{f}}{d\theta}=-\left[\tilde{\lambda}(1-\theta)\upsilon_{f}^{\sigma}\right]^{\frac{1}{\sigma+\theta}}\left[\frac{1}{(1-\theta)(\sigma+\theta)}+\frac{\log\left(\tilde{\lambda}(1-\theta)\upsilon_{f}^{\sigma}\right)}{(\sigma+\theta)^{2}}\right]$
(C.35)

Next, differentiating the compensated tax revenue function (C.33) by taxable
income,

$\frac{\partial
T}{\partial\tilde{y}_{m}}=1-\tilde{\lambda}(1-\theta)\tilde{y}_{m}^{-\theta}=1-\tilde{\lambda}^{\frac{\sigma}{\sigma+\theta}}(1-\theta)^{\frac{\sigma}{\sigma+\theta}}\upsilon_{m}^{-\frac{\sigma\theta}{\sigma+\theta}}$
(C.36) $\frac{\partial
T}{\partial\tilde{y}_{f}}=1-\tilde{\lambda}(1-\theta)\tilde{y}_{f}^{-\theta}=1-\tilde{\lambda}^{\frac{\sigma}{\sigma+\theta}}(1-\theta)^{\frac{\sigma}{\sigma+\theta}}\upsilon_{f}^{-\frac{\sigma\theta}{\sigma+\theta}}$
(C.37)

Using (C.34)-(C.35) and (C.36)-(C.37), I obtain the expression for marginal
efficiency loss for $\left(\upsilon_{m},\upsilon_{f}\right)$-couple under HSV
tax and transfer function:

$\frac{dD_{sep}\left(\upsilon_{m},\upsilon_{f}\right)}{d\theta}=-\left[\frac{\partial
T}{\partial\tilde{y}_{m}}\cdot\frac{d\tilde{y}_{m}}{d\theta}+\frac{\partial
T}{\partial\tilde{y}_{f}}\cdot\frac{d\tilde{y}_{f}}{d\theta}\right]=\\\
\left[1-\tilde{\lambda}^{\frac{\sigma}{\sigma+\theta}}(1-\theta)^{\frac{\sigma}{\sigma+\theta}}\upsilon_{m}^{-\frac{\sigma\theta}{\sigma+\theta}}\right]\frac{\left[\tilde{\lambda}(1-\theta)^{1-\sigma-\theta}\upsilon_{m}^{\sigma}\right]^{\frac{1}{\sigma+\theta}}}{\sigma+\theta}\left[1+\frac{(1-\theta)\log\left(\tilde{\lambda}(1-\theta)\upsilon_{m}^{\sigma}\right)}{\sigma+\theta}\right]+\\\
\left[1-\tilde{\lambda}^{\frac{\sigma}{\sigma+\theta}}(1-\theta)^{\frac{\sigma}{\sigma+\theta}}\upsilon_{f}^{-\frac{\sigma\theta}{\sigma+\theta}}\right]\frac{\left[\tilde{\lambda}(1-\theta)^{1-\sigma-\theta}\upsilon_{f}^{\sigma}\right]^{\frac{1}{\sigma+\theta}}}{\sigma+\theta}\left[1+\frac{(1-\theta)\log\left(\tilde{\lambda}(1-\theta)\upsilon_{f}^{\sigma}\right)}{\sigma+\theta}\right]$
(C.38)

Now I turn to the linearized program. Given the values of parameters
$\left(\theta,\tilde{\lambda},\sigma,\upsilon_{m},\upsilon_{f}\right)$, denote
the solution to the couple’s problem by
$\left(c^{*},y^{*}_{m},y^{*}_{f}\right)$. Linearizing the budget constraint
(C.27) around this point, I obtain

$c=\tilde{\lambda}(1-\theta)\left(y_{m}^{*}\right)^{-\theta}y_{m}+\tilde{\lambda}(1-\theta)\left(y_{f}^{*}\right)^{-\theta}y_{f}+T^{*}$
(C.39)

where
$T^{*}=\tilde{\lambda}\theta\left(y_{m}^{*}\right)^{1-\theta}+\tilde{\lambda}\theta\left(y_{f}^{*}\right)^{1-\theta}+c^{*}$.

Next, I plug the linearized budget constraint (C.39) into the objective
function (C.26) and obtain the following first-order conditions:

$\left(\frac{y_{m}}{\upsilon_{m}}\right)^{\sigma}=\tilde{\lambda}(1-\theta)\left(y_{m}^{*}\right)^{-\theta}$
(C.40)
$\left(\frac{y_{f}}{\upsilon_{f}}\right)^{\sigma}=\tilde{\lambda}(1-\theta)\left(y_{f}^{*}\right)^{-\theta}$
(C.41)

Optimal taxable incomes in the problem with a linearized budget constraint are
given by

$\tilde{y}^{L}_{m}=\tilde{\lambda}^{\frac{1}{\sigma}}(1-\theta)^{\frac{1}{\sigma}}\left(y_{m}^{*}\right)^{-\frac{\theta}{\sigma}}\upsilon_{m}$
(C.42)
$\tilde{y}^{L}_{f}=\tilde{\lambda}^{\frac{1}{\sigma}}(1-\theta)^{\frac{1}{\sigma}}\left(y_{f}^{*}\right)^{-\frac{\theta}{\sigma}}\upsilon_{f}$
(C.43)

Differentiating the compensated taxable income functions $\tilde{y}^{L}_{m}$
and $\tilde{y}^{L}_{f}$ with respect to the tax progressivity parameter
$\theta$, I obtain

$\frac{d\tilde{y}^{L}_{m}}{d\theta}=-\left[\tilde{\lambda}(1-\theta)^{1-\sigma}\left(y_{m}^{*}\right)^{-\theta}\right]^{\frac{1}{\sigma}}\left[\frac{1}{\sigma}+\frac{(1-\theta)\log
y_{m}^{*}}{\sigma}\right]\upsilon_{m}$ (C.44)
$\frac{d\tilde{y}^{L}_{f}}{d\theta}=-\left[\tilde{\lambda}(1-\theta)^{1-\sigma}\left(y_{f}^{*}\right)^{-\theta}\right]^{\frac{1}{\sigma}}\left[\frac{1}{\sigma}+\frac{(1-\theta)\log
y_{f}^{*}}{\sigma}\right]\upsilon_{f}$ (C.45)

Plugging the optimal taxable income, (C.30) and (C.31), into these
expressions, I get

$\frac{\partial\tilde{y}^{L}_{m}}{\partial\theta}=-\left[\tilde{\lambda}(1-\theta)^{1-\sigma-\theta}\upsilon_{m}^{\sigma}\right]^{\frac{1}{\sigma+\theta}}\left[\frac{1}{\sigma}+\frac{(1-\theta)\log\left(\tilde{\lambda}(1-\theta)\upsilon_{m}^{\sigma}\right)}{\sigma(\sigma+\theta)}\right]$
(C.46)
$\frac{\partial\tilde{y}^{L}_{f}}{\partial\theta}=-\left[\tilde{\lambda}(1-\theta)^{1-\sigma-\theta}\upsilon_{f}^{\sigma}\right]^{\frac{1}{\sigma+\theta}}\left[\frac{1}{\sigma}+\frac{(1-\theta)\log\left(\tilde{\lambda}(1-\theta)\upsilon_{f}^{\sigma}\right)}{\sigma(\sigma+\theta)}\right]$
(C.47)

By construction, optimal taxable income in the problems with nonlinear and
linearized tax functions coincide, i.e. $\tilde{y}_{m}=\tilde{y}^{L}_{m}$ and
$\tilde{y}_{f}=\tilde{y}^{L}_{f}$. Therefore, $\partial
T/\partial\tilde{y}_{m}=\partial T/\partial\tilde{y}^{L}_{m}$ and $\partial
T/\partial\tilde{y}_{f}=\partial T/\partial\tilde{y}^{L}_{f}$. Using
(C.36)-(C.37) and (C.46)-(C.47), I obtain marginal deadweight loss for
$\left(\upsilon_{m},\upsilon_{f}\right)$-couple in the problem with a
linearized HSV tax function:

$\frac{dD^{L}_{sep}\left(\upsilon_{m},\upsilon_{f}\right)}{d\theta}=-\left[\frac{\partial
T}{\partial\tilde{y}^{L}_{m}}\cdot\frac{d\tilde{y}^{L}_{m}}{d\theta}+\frac{\partial
T}{\partial\tilde{y}^{L}_{f}}\cdot\frac{d\tilde{y}^{L}_{f}}{d\theta}\right]=\\\
\left[1-\tilde{\lambda}^{\frac{\sigma}{\sigma+\theta}}(1-\theta)^{\frac{\sigma}{\sigma+\theta}}\upsilon_{m}^{-\frac{\sigma\theta}{\sigma+\theta}}\right]\frac{\left[\tilde{\lambda}(1-\theta)^{1-\sigma-\theta}\upsilon_{m}^{\sigma}\right]^{\frac{1}{\sigma+\theta}}}{\sigma}\left[1+\frac{(1-\theta)\log\left(\tilde{\lambda}(1-\theta)\upsilon_{m}^{\sigma}\right)}{\sigma+\theta}\right]+\\\
\left[1-\tilde{\lambda}^{\frac{\sigma}{\sigma+\theta}}(1-\theta)^{\frac{\sigma}{\sigma+\theta}}\upsilon_{f}^{-\frac{\sigma\theta}{\sigma+\theta}}\right]\frac{\left[\tilde{\lambda}(1-\theta)^{1-\sigma-\theta}\upsilon_{f}^{\sigma}\right]^{\frac{1}{\sigma+\theta}}}{\sigma}\left[1+\frac{(1-\theta)\log\left(\tilde{\lambda}(1-\theta)\upsilon_{f}^{\sigma}\right)}{\sigma+\theta}\right]$
(C.48)

Finally, using the government budget constraint, I solve for $\tilde{\lambda}$
as a function of policy parameters $(\theta,g)$ and primitives of the economy.
The government budget constraint under separate taxation of spousal incomes
takes the following form:

$g\int\left(y_{m}+y_{f}\right)d\varGamma\left(\upsilon_{m},\upsilon_{f}\right)=\int\left(y_{m}+y_{f}\right)d\varGamma\left(\upsilon_{m},\upsilon_{f}\right)-\tilde{\lambda}\int\left(y_{m}^{1-\theta}+y_{f}^{1-\theta}\right)d\varGamma\left(\upsilon_{m},\upsilon_{f}\right)$
(C.49)

Solving for $\tilde{\lambda}$, I obtain

$\tilde{\lambda}=\frac{(1-g)\int\left(y_{m}+y_{f}\right)d\varGamma\left(\upsilon_{m},\upsilon_{f}\right)}{\int\left(y_{m}^{1-\theta}+y_{f}^{1-\theta}\right)d\varGamma\left(\upsilon_{m},\upsilon_{f}\right)}$
(C.50)

Finally, plugging (C.30) and (C.31), I get the expression for the equilibrium
value of $\tilde{\lambda}$ under separate taxation of spouses:

$\tilde{\lambda}=(1-g)^{\frac{\sigma+\theta}{\sigma}}(1-\theta)^{\frac{\theta}{\sigma}}\left[\frac{\int\left(\upsilon_{m}^{\frac{\sigma}{\sigma+\theta}}+\upsilon_{f}^{\frac{\sigma}{\sigma+\theta}}\right)d\varGamma\left(\upsilon_{m},\upsilon_{f}\right)}{\int\left(\upsilon_{m}^{\frac{\sigma(1-\theta)}{\sigma+\theta}}+\upsilon_{f}^{\frac{\sigma(1-\theta)}{\sigma+\theta}}\right)d\varGamma\left(\upsilon_{m},\upsilon_{f}\right)}\right]^{\frac{\sigma+\theta}{\sigma}}$
(C.51)

Note that, by construction, $\tilde{y}_{m}=\tilde{y}^{L}_{m}$ and
$\tilde{y}_{f}=\tilde{y}^{L}_{f}$, and hence the values of $\tilde{\lambda}$
in the original and linearized programs coincide. This completes the
derivation of the expressions for reform-induced efficiency loss in the case
of separate taxation of spousal incomes.

Overall, this completes the proof of Proposition C.1. $\blacksquare$

### Proof of Proposition 3 (Linearization Bias with HSV Tax Function)

Joint Taxation of Spouses

By definition, the linearization bias is given by

$\Delta_{joint}=\frac{\frac{dD^{L}_{joint}}{d\theta}-\frac{dD_{joint}}{d\theta}}{\frac{dD_{joint}}{d\theta}}=\frac{\int\frac{dD^{L}_{joint}\left(\upsilon_{m},\upsilon_{f}\right)}{d\theta}d\varGamma\left(\upsilon_{m},\upsilon_{f}\right)-\int\frac{dD_{joint}\left(\upsilon_{m},\upsilon_{f}\right)}{d\theta}d\varGamma\left(\upsilon_{m},\upsilon_{f}\right)}{\int\frac{dD_{joint}\left(\upsilon_{m},\upsilon_{f}\right)}{d\theta}d\varGamma\left(\upsilon_{m},\upsilon_{f}\right)}$

Plugging (C.12) and (C.22) from the first part of Proposition C.1, I obtain

$\Delta_{joint}=\frac{\left(\frac{1}{\sigma}-\frac{1}{\sigma+\theta}\right)\int\left[1-\lambda^{\frac{\sigma}{\sigma+\theta}}(1-\theta)^{\frac{\sigma}{\sigma+\theta}}\left(\upsilon_{m}+\upsilon_{f}\right)^{-\frac{\sigma\theta}{\sigma+\theta}}\right]\left[\lambda(1-\theta)^{1-\sigma-\theta}\left(\upsilon_{m}+\upsilon_{f}\right)^{\sigma}\right]^{\frac{1}{\sigma+\theta}}\left[1+\frac{(1-\theta)\log\left(\lambda(1-\theta)\left(\upsilon_{m}+\upsilon_{f}\right)^{\sigma}\right)}{\sigma+\theta}\right]d\varGamma\left(\upsilon_{m},\upsilon_{f}\right)}{\frac{1}{\sigma+\theta}\int\left[1-\lambda^{\frac{\sigma}{\sigma+\theta}}(1-\theta)^{\frac{\sigma}{\sigma+\theta}}\left(\upsilon_{m}+\upsilon_{f}\right)^{-\frac{\sigma\theta}{\sigma+\theta}}\right]\left[\lambda(1-\theta)^{1-\sigma-\theta}\left(\upsilon_{m}+\upsilon_{f}\right)^{\sigma}\right]^{\frac{1}{\sigma+\theta}}\left[1+\frac{(1-\theta)\log\left(\lambda(1-\theta)\left(\upsilon_{m}+\upsilon_{f}\right)^{\sigma}\right)}{\sigma+\theta}\right]d\varGamma\left(\upsilon_{m},\upsilon_{f}\right)}$

Finally, simplifying, I find that the linearization bias is given by the ratio
between the tax progressivity parameter $\theta$ and the inverse elasticity of
taxable income $\sigma$:

$\Delta_{joint}=\frac{\frac{1}{\sigma}-\frac{1}{\sigma+\theta}}{\frac{1}{\sigma+\theta}}=\frac{\theta}{\sigma}$
(C.52)

Separate Taxation of Spouses

The linearization bias is given by

$\Delta_{sep}=\frac{\frac{dD^{L}_{sep}}{d\theta}-\frac{dD_{sep}}{d\theta}}{\frac{dD_{sep}}{d\theta}}=\frac{\int\frac{dD^{L}_{sep}\left(\upsilon_{m},\upsilon_{f}\right)}{d\theta}d\varGamma\left(\upsilon_{m},\upsilon_{f}\right)-\int\frac{dD_{sep}\left(\upsilon_{m},\upsilon_{f}\right)}{d\theta}d\varGamma\left(\upsilon_{m},\upsilon_{f}\right)}{\int\frac{dD_{sep}\left(\upsilon_{m},\upsilon_{f}\right)}{d\theta}d\varGamma\left(\upsilon_{m},\upsilon_{f}\right)}$

Plugging (C.38) and (C.48) from the second part of Proposition C.1, I obtain

$\Delta_{sep}=\frac{\left(\frac{1}{\sigma}-\frac{1}{\sigma+\theta}\right)\int\sum_{j=m,f}\left[1-\tilde{\lambda}^{\frac{\sigma}{\sigma+\theta}}(1-\theta)^{\frac{\sigma}{\sigma+\theta}}\upsilon_{j}^{-\frac{\sigma\theta}{\sigma+\theta}}\right]\left[\tilde{\lambda}(1-\theta)^{1-\sigma-\theta}\upsilon_{j}^{\sigma}\right]^{\frac{1}{\sigma+\theta}}\left[1+\frac{(1-\theta)\log\left(\tilde{\lambda}(1-\theta)\upsilon_{j}^{\sigma}\right)}{\sigma+\theta}\right]d\varGamma\left(\upsilon_{m},\upsilon_{f}\right)}{\frac{1}{\sigma+\theta}\int\sum_{j=m,f}\left[1-\tilde{\lambda}^{\frac{\sigma}{\sigma+\theta}}(1-\theta)^{\frac{\sigma}{\sigma+\theta}}\upsilon_{j}^{-\frac{\sigma\theta}{\sigma+\theta}}\right]\left[\tilde{\lambda}(1-\theta)^{1-\sigma-\theta}\upsilon_{j}^{\sigma}\right]^{\frac{1}{\sigma+\theta}}\left[1+\frac{(1-\theta)\log\left(\tilde{\lambda}(1-\theta)\upsilon_{j}^{\sigma}\right)}{\sigma+\theta}\right]d\varGamma\left(\upsilon_{m},\upsilon_{f}\right)}$

Finally, simplifying, I find that the linearization bias coincides with one I
obtain under joint taxation of couples:

$\Delta_{sep}=\frac{\frac{1}{\sigma}-\frac{1}{\sigma+\theta}}{\frac{1}{\sigma+\theta}}=\frac{\theta}{\sigma}$
(C.53)

This completes the proof of Proposition 3. $\blacksquare$

## Additional Figures and Tables

Figure A.1: Annual working hours of married women with working husbands in the
United States

Notes: Data is from the Annual Social and Economic Supplement of the Current
Population Survey. The sample includes married women aged 25-54 with working
husbands.

(a) TRA 1986 reform

(b) OBRA 1993 reform

(c) EGTRRA 2001 reform

(d) TCJA 2017 reform

Figure A.2: Pre-reform and post-reform U.S. federal income tax brackets and statutory marginal tax rates for married couples filing jointly Table A.1: Summary statistics, 1986 and 1992 | 1986 | 1992
---|---|---
| Mean | Median | St. Dev. | Mean | Median | St. Dev.
Males |  |  |  |  |  |
Age | 38.94 | 38 | 7.88 | 39.48 | 39 | 7.61
White | 0.896 | 1 | 0.305 | 0.892 | 1 | 0.311
College degree | 0.291 | 0 | 0.454 | 0.311 | 0 | 0.463
Annual hours | 2201 | 2080 | 588 | 2217 | 2080 | 606
Earnings (2012 USD) | 52873 | 47893 | 30218 | 53919 | 47610 | 33521
Females |  |  |  |  |  |
Age | 36.66 | 36 | 7.55 | 37.47 | 37 | 7.37
White | 0.895 | 1 | 0.306 | 0.891 | 1 | 0.312
College degree | 0.211 | 0 | 0.408 | 0.259 | 0 | 0.438
Employment | 0.732 | 1 | 0.443 | 0.764 | 1 | 0.425
Annual hours | 1214 | 1400 | 940 | 1330 | 1664 | 939
Earnings (2012 USD) | 25946 | 22104 | 19507 | 29740 | 25293 | 21906
Number of children | 1.62 | 2 | 1.22 | 1.54 | 2 | 1.19
Number of children under 6 | 0.50 | 0 | 0.78 | 0.49 | 0 | 0.77
Female — secondary earner | 0.834 | 1 | 0.372 | 0.788 | 1 | 0.409
Number of observations | 17127 | 18032

Notes: Data is from the Annual Social and Economic Supplement of the Current
Population Survey. The sample includes married couples aged 25-54 with working
husbands. Secondary earner is defined as the person with the lowest income
among two spouses. Details about sample selection are given in the main text.
To calculate the summary statistics, I use the CPS ASEC weights.

Table A.2: Summary statistics, 2000 and 2017 | 2000 | 2017
---|---|---
| Mean | Median | St. Dev. | Mean | Median | St. Dev.
Males |  |  |  |  |  |
Age | 40.63 | 41 | 7.63 | 40.76 | 41 | 7.77
White | 0.865 | 1 | 0.341 | 0.812 | 1 | 0.391
College degree | 0.351 | 0 | 0.477 | 0.440 | 0 | 0.496
Annual hours | 2294 | 2080 | 558 | 2229 | 2080 | 532
Earnings (2012 USD) | 72918 | 53688 | 74811 | 76318 | 56644 | 81251
Females |  |  |  |  |  |
Age | 38.78 | 39 | 7.57 | 38.96 | 39 | 7.78
White | 0.862 | 1 | 0.345 | 0.804 | 1 | 0.397
College degree | 0.324 | 0 | 0.468 | 0.493 | 0 | 0.500
Employment rate | 0.777 | 1 | 0.416 | 0.747 | 1 | 0.435
Annual hours | 1393 | 1820 | 947 | 1388 | 1872 | 971
Earnings (2012 USD) | 37659 | 31332 | 37063 | 49817 | 37763 | 54504
Number of children | 1.55 | 2 | 1.23 | 1.61 | 2 | 1.27
Number of children under 6 | 0.46 | 0 | 0.76 | 0.51 | 0 | 0.79
Female — secondary earner | 0.775 | 1 | 0.417 | 0.720 | 1 | 0.449
Number of observations | 26883 | 17415

Notes: Data is from the Annual Social and Economic Supplement of the Current
Population Survey. The sample includes married couples aged 25-54 with working
husbands. Secondary earner is defined as the person with the lowest income
among two spouses. Details about sample selection are given in the main text.
To calculate the summary statistics, I use the CPS ASEC weights.

## Online Appendix (Not for Publication)

##  Appendix A NBER TAXSIM Inputs

To calculate the tax liabilities, the Internet NBER TAXSIM uses 32 input
variables. As described in the text, I set most of them to zero because I do
not model such things as childcare, capital income, housing, etc. Below I
provide the full list of input variables with the details on each field.

  1. 1.

taxsimid: Individual ID.

  2. 2.

year: Tax year.

  3. 3.

state = 23 (Michigan): State.

  4. 4.

mstat = 2 (married filing jointly): Marital status.

  5. 5.

page: Age of primary taxpayer.

  6. 6.

sage: Age of spouse.

  7. 7.

depx = 2: Number of dependents.

  8. 8.

dep13 = 2: Number of children under 13.

  9. 9.

dep17 = 2: Number of children under 17.

  10. 10.

dep18 = 2: Number of qualifying children for EITC.

  11. 11.

pwages: Wage and salary income of primary taxpayer (including self-
employment).

  12. 12.

swages: Wage and salary income of spouse (including self-employment).

  13. 13.

dividends = 0: Dividend income.

  14. 14.

intrec = 0: Interest received.

  15. 15.

stcg = 0: Short term capital gains or losses.

  16. 16.

ltcg = 0: Long term capital gains or losses.

  17. 17.

otherprop = 0: Other property income subject to Net Investment Income Tax
(NIIT).

  18. 18.

nonprop = 0: Other non-property income not subject to Medicare NIIT.

  19. 19.

pensions = 0: Taxable pensions and IRA distributions.

  20. 20.

gssi = 0: Gross Social Security benefits.

  21. 21.

ui = 0: Unemployment compensation received.

  22. 22.

transfers = 0: Other non-taxable transfer income.

  23. 23.

rentpaid = 0: Rent paid.

  24. 24.

proptax = 0: Real estate taxes paid.

  25. 25.

otheritem = 0: Other itemized deductions that are a preference for the
Alternative Minimum Tax (AMT).

  26. 26.

childcare = 0: Child care expenses.

  27. 27.

mortgage = 0: Deductions not included in otheritem and not a preference for
the AMT.

  28. 28.

scorp = 0: Active S-Corp income.

  29. 29.

pbusinc = 0: Primary taxpayer’s Qualified Business Income (QBI) subject to a
preferential rate without phaseout.

  30. 30.

pprofinc = 0: Primary taxpayer’s Specialized Service Trade or Business service
(SSTB) with a preferential rate subject to claw-back.

  31. 31.

sbusinc = 0: Spouse’s QBI.

  32. 32.

sprofinc = 0: Spouse’s SSTB.

##  Appendix B Supplementary Tables

Table OA.1: EITC parameters for U.S. married couples filing jointly, 1986-2018 Year |  | Eligible
---
Children
| Phase-In
---
Rate, %
| First
---
Kink, $
| Maximum
---
Credit, $
| Second
---
Kink, $
| Phase-Out
---
Rate, %
| Exhaustion
---
Point, $
1986 | any | 11 | 5000 | 550 | 6500 | 12.22 | 11000
1987 | any | 14 | 6080 | 851 | 6920 | 10 | 15432
1988 | any | 14 | 6240 | 874 | 9840 | 10 | 18576
1989 | any | 14 | 6500 | 910 | 10240 | 10 | 19340
1990 | any | 14 | 6810 | 953 | 10730 | 10 | 20264
1991 |  | 1
---
2+
| 16.7
---
17.3
| 7140
---
7140
| 1192
---
1235
| 11250
---
11250
| 11.93
---
12.36
| 21250
---
21250
1992 |  | 1
---
2+
| 17.6
---
18.4
| 7520
---
7520
| 1324
---
1384
| 11840
---
11840
| 12.57
---
13.14
| 22370
---
22370
1993 |  | 1
---
2+
| 18.5
---
19.5
| 7750
---
7750
| 1434
---
1511
| 12200
---
12200
| 13.21
---
13.93
| 23050
---
23050
1994 |  | 0
---
1
2+
| 7.65
---
26.3
30
| 4000
---
7750
8425
| 306
---
2038
2528
| 5000
---
11000
11000
| 7.65
---
15.98
17.68
| 9000
---
23755
25296
1995 |  | 0
---
1
2+
| 7.65
---
34
36
| 4100
---
6160
8640
| 314
---
2094
3110
| 5130
---
11290
11290
| 7.65
---
15.98
20.22
| 9230
---
24396
26673
1996 |  | 0
---
1
2+
| 7.65
---
34
40
| 4220
---
6330
8890
| 323
---
2152
3556
| 5280
---
11610
11610
| 7.65
---
15.98
21.06
| 9500
---
25078
28495
1997 |  | 0
---
1
2+
| 7.65
---
34
40
| 4340
---
6500
9140
| 332
---
2210
3656
| 5430
---
11930
11930
| 7.65
---
15.98
21.06
| 9770
---
25750
29290
1998 |  | 0
---
1
2+
| 7.65
---
34
40
| 4460
---
6680
9390
| 341
---
2271
3756
| 5570
---
12260
12260
| 7.65
---
15.98
21.06
| 10030
---
26473
30095
1999 |  | 0
---
1
2+
| 7.65
---
34
40
| 4530
---
6800
9540
| 347
---
2312
3816
| 5670
---
12460
12460
| 7.65
---
15.98
21.06
| 10200
---
26928
30580
2000 |  | 0
---
1
2+
| 7.65
---
34
40
| 4610
---
6920
9720
| 353
---
2353
3888
| 5770
---
12690
12690
| 7.65
---
15.98
21.06
| 10380
---
27413
31152
2001 |  | 0
---
1
2+
| 7.65
---
34
40
| 4760
---
7140
10020
| 364
---
2428
4008
| 5950
---
13090
13090
| 7.65
---
15.98
21.06
| 10710
---
28281
32121
2002 |  | 0
---
1
2+
| 7.65
---
34
40
| 4910
---
7370
10350
| 376
---
2506
4140
| 7150
---
14520
14520
| 7.65
---
15.98
21.06
| 12060
---
30201
34178
2003 |  | 0
---
1
2+
| 7.65
---
34
40
| 4990
---
7490
10510
| 382
---
2547
4204
| 7240
---
14730
14730
| 7.65
---
15.98
21.06
| 12230
---
30666
34692
2004 |  | 0
---
1
2+
| 7.65
---
34
40
| 5100
---
7660
10750
| 390
---
2604
4300
| 7390
---
15040
15040
| 7.65
---
15.98
21.06
| 12490
---
31338
35458
2005 |  | 0
---
1
2+
| 7.65
---
34
40
| 5220
---
7830
11000
| 399
---
2662
4400
| 8530
---
16370
16370
| 7.65
---
15.98
21.06
| 13750
---
33030
37263
2006 |  | 0
---
1
2+
| 7.65
---
34
40
| 5380
---
8080
11340
| 412
---
2747
4536
| 8740
---
16810
16810
| 7.65
---
15.98
21.06
| 14120
---
34001
38348
2007 |  | 0
---
1
2+
| 7.65
---
34
40
| 5590
---
8390
11790
| 428
---
2853
4716
| 9000
---
17390
17390
| 7.65
---
15.98
21.06
| 14590
---
35241
39783
2008 |  | 0
---
1
2+
| 7.65
---
34
40
| 5720
---
8580
12060
| 438
---
2917
4824
| 10160
---
18740
18740
| 7.65
---
15.98
21.06
| 15880
---
36995
41646
2009 |  | 0
---
1
2
3+
| 7.65
---
34
40
45
| 5970
---
8950
12570
12570
| 457
---
3043
5028
5657
| 12470
---
21420
21420
21420
| 7.65
---
15.98
21.06
21.06
| 18440
---
40463
45295
48279
2010 |  | 0
---
1
2
3+
| 7.65
---
34
40
45
| 5980
---
8970
12590
12590
| 457
---
3050
5036
5666
| 12490
---
21460
21460
21460
| 7.65
---
15.98
21.06
21.06
| 18470
---
40545
45373
48362
2011 |  | 0
---
1
2
3+
| 7.65
---
34
40
45
| 6070
---
9100
12780
12780
| 464
---
3094
5112
5751
| 12670
---
21770
21770
21770
| 7.65
---
15.98
21.06
21.06
| 15740
---
41132
46044
49078
2012 |  | 0
---
1
2
3+
| 7.65
---
34
40
45
| 6210
---
9320
13090
13090
| 475
---
3169
5236
5891
| 12980
---
22300
22300
22300
| 7.65
---
15.98
21.06
21.06
| 19190
---
42130
47162
50270
2013 |  | 0
---
1
2
3+
| 7.65
---
34
40
45
| 6370
---
9560
13430
13430
| 487
---
3250
5372
6044
| 13310
---
22870
22870
22870
| 7.65
---
15.98
21.06
21.06
| 19680
---
43210
48378
51567
2014 |  | 0
---
1
2
3+
| 7.65
---
34
40
45
| 6480
---
9720
13650
13650
| 496
---
3305
5460
6143
| 13540
---
23260
23260
23260
| 7.65
---
15.98
21.06
21.06
| 20020
---
43941
49186
52427
2015 |  | 0
---
1
2
3+
| 7.65
---
34
40
45
| 6580
---
9880
13870
13870
| 503
---
3359
5548
6242
| 13760
---
23630
23630
23630
| 7.65
---
15.98
21.06
21.06
| 20340
---
44651
49974
53267
2016 |  | 0
---
1
2
3+
| 7.65
---
34
40
45
| 6610
---
9920
13931
13930
| 506
---
3373
5572
6269
| 13820
---
23740
23740
23740
| 7.65
---
15.98
21.06
21.06
| 20430
---
44846
50198
53505
2017 |  | 0
---
1
2
3+
| 7.65
---
34
40
45
| 6670
---
10000
14040
14040
| 510
---
3400
5616
6318
| 13930
---
23930
23930
23930
| 7.65
---
15.98
21.06
21.06
| 20600
---
45207
50597
53930
2018 |  | 0
---
1
2
3+
| 7.65
---
34
40
45
| 6780
---
10180
14290
14290
| 519
---
3461
5716
6431
| 14170
---
24350
24350
24350
| 7.65
---
15.98
21.06
21.06
| 20950
---
46010
51492
54884

Notes: This table shows the federal Earned Income Tax Credit parameters by
family size for married couples filing jointly. Eligible children are under
age 19 (under 24 if a full-time student) or permanently disabled and must
reside with the taxpayer for more than half a year. Since 2002, the values of
the second kink and exhaustion point were increased for married taxpayers
filing jointly relative to taxpayers filing as single or the head of
household. The phase-in rate is defined as the increase in the tax credit for
an additional dollar of income. The first kink point corresponds to minimum
income that is needed for maximizing the size of tax credit. The second kink
point corresponds to maximum income allowed before the phasing-out region. The
phase-out rate is defined as the reduction in the tax credit for an additional
dollar of income above the second kink point. The exhaustion point corresponds
to income at which the Earned Income Tax Credit is completely phased out.

Table OA.2: Standard deductions and personal exemptions for U.S. married couples filing jointly Year | Standard Deduction | Personal Exemption
---|---|---
1986 | 3670 | 1080
1987 | 3760 | 1900
1988 | 5000 | 1950
1989 | 5200 | 2000
1990 | 5450 | 2050
1991 | 5700 | 2150
1992 | 6000 | 2300
1993 | 6200 | 2350
1994 | 6350 | 2450
1995 | 6550 | 2500
1996 | 6700 | 2550
1997 | 6900 | 2650
1998 | 7100 | 2700
1999 | 7200 | 2750
2000 | 7350 | 2800
2001 | 7600 | 2900
2002 | 7850 | 3000
2003 | 9500 | 3050
2004 | 9700 | 3100
2005 | 10000 | 3200
2006 | 10300 | 3300
2007 | 10700 | 3400
2008 | 10900 | 3500
2009 | 11400 | 3650
2010 | 11400 | 3650
2011 | 11600 | 3700
2012 | 11900 | 3800
2013 | 12200 | 3900
2014 | 12400 | 3950
2015 | 12600 | 4000
2016 | 12600 | 4050
2017 | 12700 | 4050
2018 | 24000 | 0

Notes: The Tax Cuts and Jobs Act of 2017 eliminated personal exemptions for
tax years 2018-2025.