# Welfare Effects of Labor Income Tax Changes
on Married Couples: A Sufficient Statistics Approach

Egor Malkov111 University of Minnesota and the Federal Reserve Bank of
Minneapolis. E-mail<EMAIL_ADDRESS>I thank Fatih Guvenen, Larry Jones, and
Fabrizio Perri for their valuable advice and constant support. I thank Carlos
E. da Costa, Diane Lim, and Xincheng Qiu for insightful discussions. I thank
Nezih Guner, Andreas Haufler, Julie Hotchkiss, Jonas Loebbing, Andreas Peichl,
Dominik Sachs, Minchul Yum, Daniel Weishaar, and the participants of the
European Economic Association Congress 2020, VMACS Junior Conference, National
Tax Association Annual Conference 2020, Midwest Economics Association 2021
Annual Meeting, SEHO 2021 Meeting, ESPE 2021 Conference, COMPIE 2021
Conference, Public Economics Seminar at the LMU Munich, NIPFP-IIPF
International Conference on Public Finance, 2021 ECINEQ Meeting, China Meeting
of the Econometric Society, and Brazilian Meeting in Family and Gender
Economics for fruitful comments. The views expressed herein are those of the
author and not necessarily those of the Federal Reserve Bank of Minneapolis or
the Federal Reserve System.

(This version: July 2021)

This paper develops a framework for assessing the welfare effects of labor
income tax changes on married couples. I build a static model of couples’
labor supply that features both intensive and extensive margins and derive a
tractable expression that delivers a transparent understanding of how labor
supply responses, policy parameters, and income distribution affect the
reform-induced welfare gains. Using this formula, I conduct a comparative
welfare analysis of four tax reforms implemented in the United States over the
last four decades, namely the Tax Reform Act of 1986, the Omnibus Budget
Reconciliation Act of 1993, the Economic Growth and Tax Relief Reconciliation
Act of 2001, and the Tax Cuts and Jobs Act of 2017. I find that these reforms
created welfare gains ranging from -0.16 to 0.62 percent of aggregate labor
income. A sizable part of the gains is generated by the labor force
participation responses of women. Despite three reforms resulted in aggregate
welfare gains, I show that each reform created both winners and losers.
Furthermore, I uncover two patterns in the relationship between welfare gains
and couples’ labor income. In particular, the reforms of 1986 and 2017 display
a monotonically increasing relationship, while the other two reforms
demonstrate a U-shaped pattern. Finally, I characterize the bias in welfare
gains resulting from the assumption about a linear tax function. I consider a
reform that changes tax progressivity and show that the linearization bias is
given by the ratio between the tax progressivity parameter and the inverse
elasticity of taxable income. Quantitatively, it means that linearization
overestimates the welfare effects of the U.S. tax reforms by 3.6-18.1%.

JEL: D60, E62, E65, H31, J22.

Keywords: Taxation of Couples, Tax Reforms, Welfare Analysis, Labor Supply,
Sufficient Statistics, Linearization Bias.

##  1 Introduction

What are the welfare effects of tax reforms on married couples? How are the
welfare gains and losses distributed among them? The answers to these
questions are of crucial importance for both academic economists and
policymakers for several reasons. First, the scope is significant. Married
couples constitute a sizable share of the population (e.g., they account for
almost a half of all the U.S. households in 2019) and taxpayers (e.g., the
number of tax returns of married couples filing jointly constitutes more than
a third of all tax returns in the United States). Second, positive assortative
mating, when similarly educated individuals tend to marry each other, is
considered as one of the driving forces of between-household inequality (Dupuy
and Weber, 2021). Therefore, it is crucial to know who benefits and who loses
from the redistributive policies such as income taxation. Finally, the tax and
transfer systems that feature jointness, such as in Germany and the United
States, create substantial disincentive effects for the married women’s labor
supply (Bick and Fuchs-Schündeln, 2017a; Holter et al., 2019). Under joint
taxation of spousal incomes, the marginal tax rate on the first dollar earned
by the secondary (lower income) earner is equal to the marginal tax rate on
the last dollar earned by the primary (higher income) earner. Figure 1
illustrates the last point by showing the participation tax rates (change in
household’s tax liability divided by woman’s earnings when she starts working)
for married and single women in the United States. Except for the marital
status, these two women are otherwise identical. Clearly, a married woman
faces a significantly higher tax rate, when she starts working, than a single
one. A tax reform that results in lower participation tax rates, as shown by
the difference between dashed and solid lines, can enhance employment of the
married women. Overall, studying the positive aspects of income taxation of
couples is critically important because it is closely related to a vast array
of topics including between- and within-household inequality, female labor
supply, marriage decisions, and so on.

In this paper, I develop a framework for studying the welfare effects of
income tax changes on married couples. First, I build a static model of
couples’ labor supply that features both intensive and extensive margins.
Using this model, I derive a tractable expression for welfare gains, resulting
from any arbitrary small tax policy reform, as a function of several
sufficient statistics: labor supply elasticities (elasticities of hours,
cross-elasticities of spousal hours, and participation elasticity), policy
parameters (pre-reform marginal and participation tax rates and their reform-
induced changes), and labor income shares. In a transparent way, it allows
decomposing the changes in aggregate efficiency gains into the effects that
operate through labor supply responses. I use this sufficient statistics
formula to quantify the welfare effects of four tax reforms, implemented in
the United States over the last four decades. The reforms include the Tax
Reform Act of 1986 (TRA 1986), the Omnibus Budget Reconciliation Act of 1993
(OBRA 1993), the Economic Growth and Tax Relief Reconciliation Act of 2001
(EGTRRA 2001), and the Tax Cuts and Jobs Act of 2017 (TCJA 2017). To map my
expression for welfare gains to the data, I use the Current Population Survey
combined with the NBER TAXSIM calculator.

Figure 1: Participation tax rates of married and single women in the United
States

Notes: Participation tax rate is defined as the change in household’s tax
liability divided by female earnings when she starts working. The tax rates
are calculated using NBER TAXSIM and include federal, state, and FICA tax
rates. Both women aged 40, live in Minnesota, and have two children under 19.
A married woman husband’s annual earnings are fixed at the median level in
Minnesota. Individuals do not have any non-labor income. Married couple is
assumed to file jointly. Vertical line corresponds to median annual female
earnings in Minnesota (I multiply median weekly female earnings in Minnesota
by median annual working hours of women in the United States divided by 40).

Under the baseline parameterization, I estimate the efficiency gains from four
tax reforms to range from -0.16 (OBRA 1993) to 0.62 (TCJA 2017) percent of
aggregate labor income. The welfare gains per dollar spent range from 0.63 USD
(OBRA 1993) to 1.10 USD (TCJA 2017). Overall, three reforms, the TRA 1986, the
EGTRRA 2001, and the TCJA 2017, created aggregate welfare gains for married
couples. A substantial part of these gains comes from the labor force
participation responses of women. Furthermore, I also emphasize that the
spousal cross-effects of working hours (change in working hours of one spouse
resulting from the change in the net-of-tax rate of the other spouse) are
quantitatively important. Abstracting from them can lead to substantial
overestimation of efficiency gains. For example, if I abstract from the cross-
effects in the case of the TRA 1986 reform, I overestimate the welfare gains
by 34.6%. The sensitivity analysis, where I consider the range of reasonable
values of elasticities, confirms this argument.

Next, despite three out of four considered reforms generate aggregate welfare
gains, I find that each reform created winners and losers. For example, the
TRA 1986 left 12.3 percent of couples with welfare losses. Furthermore, I
study how the welfare gains vary by income and uncover two general patterns.
First, the TRA1986, the OBRA 1993 (excluding the bottom 10 percent), and the
TCJA 2017 reforms display a monotonically increasing relationship between
welfare gains and income. In other words, richer taxpayers benefited more than
poorer taxpayers. Second, the OBRA 1993 and the EGTRRA 2001 reforms
demonstrate a U-shaped pattern in the welfare gains. In the case of these
reforms, the main winners are located at the lower and upper ends of the
income distribution. Overall, my general takeaway from this part of the
analysis is that the aggregate welfare measures mask significant heterogeneity
in welfare gains.

After that, I discuss the robustness of my findings. I start with considering
alternative parameterizations of elasticities. Under the scenario that
delivers an upper bound on the efficiency gains, I find that they range from
-0.07 to 1.15 percent of aggregate labor income. Under the “lower bound”
scenario, the range from -0.25 to 0.12 percent of the aggregate labor income.
To address the concern that between the 1970s and 2000s there was a sizable
reduction in own- and cross-elasticities of married women’s labor supply (Blau
and Kahn, 2007; Heim, 2007), I also consider the parameterization where all
elasticities have reasonably high (and low) values. Next, to address the
concerns about the role of the initial income distribution and pre-reform tax
rates, I conduct two sets of counterfactual reforms. I begin with the
exercises where I apply the actual tax reforms to the counterfactual income
distributions. For example, I show that if the TRA 1986 reform were to be
applied to the 2017 income distribution, aggregate welfare gains would be 1.32
percent of aggregate labor income, or 1.14 USD per dollar spent. This exceeds
the actual welfare gains per dollar spent from the TCJA by 5.48%. Overall, I
find that counterfactual welfare gains per dollar spent do not differ by more
than 7.54% from the actual ones. In another set of exercises, I fix the income
distribution and tax law in a given pre-reform year and calculate the welfare
effects of moving to the other post-reform’s tax law. For example, I show that
moving from the pre-TRA 1986 economy to the post-EGTRRA 2001 and post-TCJA
2017 economies creates higher efficiency gains, 0.88 and 1.19 percent of
aggregate labor income, than the actual TRA 1986 (0.55 percent). However, when
I make the efficiency gains comparable, the actual TRA 1986 generates more
welfare gain per dollar than these alternative counterfactual reforms.

Finally, I address the concern that the assumption about linearity of the tax
function, commonly used in the sufficient statistics literature, may deliver
biased estimates of efficiency gains, because real tax codes feature
nonlinearities. To do so, I characterize the linearization bias, defined as
the percentage difference between reform-induced efficiency loss under true
and linearized tax and transfer functions. I assume quasilinear preferences
and use the log-linear specification $T(y)=y-\lambda y^{1-\theta}$ that
provides a good approximation of the actual tax and transfer system in the
United States (Heathcote et al., 2017). More precisely, I consider joint, i.e.
$T\left(y_{m},y_{f}\right)=T\left(y_{m}+y_{f}\right)$, and separate, i.e.
$T\left(y_{m},y_{f}\right)=T\left(y_{m}\right)+T\left(y_{m}\right)$, taxation
of spouses. Assuming a small reform that changes progressivity of the tax
system, $d\theta\approx 0$, I show that the linearization bias is given by the
ratio between the tax progressivity parameter (tax function curvature) and the
inverse elasticity of taxable income (utility curvature). Using the estimates
of these objects from the literature, I conclude that linearization biases
upward the welfare effects of the U.S. tax reforms in the range from 3.6% to
18.1%.

My paper is related to several strands of literature. First, I contribute to
the literature studying the welfare effects of tax and transfer reforms
initiated by the classic paper Harberger (1964), and further developed by
Dahlby (1998), Feldstein (1999), Kleven and Kreiner (2006), and Blomquist and
Simula (2019), among many others. On a related note, Finkelstein and Hendren
(2020) and Hendren and Sprung-Keyser (2020) emphasize the attractiveness of
the marginal value of public funds (MVPF) as a tool that allows comparing the
effects of various policies on social welfare. The paper most closely related
to mine is Eissa et al. (2008). They quantify the welfare effects of the U.S.
tax reforms on single mothers. Importantly, the results for single individuals
may be quite different from ones for married couples. First, the interactions
between spouses, captured by the cross-elasticities of working hours, are
naturally absent in a framework with singles. Second, single mothers more
likely represent the lower part of the household income distribution than
married couples, and any given tax reform may differently affect households
that belong to different income groups. Another related work by Immervoll et
al. (2009) studies the welfare effects of tax policy changes on married
couples, but the authors consider hypothetical reforms. Instead, I evaluate
the welfare gains of the actual U.S. tax reforms, including the most recent
one, the Tax Cuts and Jobs Act of 2017. Furthermore, Bar and Leukhina (2009)
investigate the impact of the U.S. tax reforms on married couples’
participation but do not allow for the intensive margin of labor supply. Next,
Hotchkiss et al. (2012) and Hotchkiss et al. (2021) evaluate the welfare
effects of the U.S. tax reforms on households, but using a different approach
than mine. Beyond that, two other related strands of literature study the
macroeconomic effects of tax reforms (Barro and Redlick, 2011; Mertens and
Ravn, 2012, 2013; Barro and Furman, 2018) and heterogeneity in the effects of
economic policy (Domeij and Heathcote, 2004; Bitler et al., 2006; Zidar,
2019).

Next, my paper is related to the literature that studies the taxation of
couples and its effects on the female labor supply. Eissa and Hoynes (2004)
find that the Earned Income Tax Credit (EITC) expansions between 1984 and 1996
reduced the total family labor supply of couples mainly through lowering the
labor force participation of married women. Next, Guner et al. (2012), using a
rich general equilibrium life-cycle model calibrated to the U.S. economy, show
that the reform replacing joint taxation to separate taxation would
substantially increase the labor supply of married women. Similarly, Bick and
Fuchs-Schündeln (2017a) emphasize that joint taxation creates significant
disincentive effects for the labor supply of married women in the United
States and Europe. In the same vein, Borella et al. (2021) show that
eliminating marriage-related taxes and old age Social Security benefits in the
United States would significantly increase the participation of married women
over their entire life cycle.

The rest of the paper is organized as follows. The model is presented in
Section 2. The data and sample selection are discussed in Section 3. The U.S.
tax reforms and construction of reform-induced tax changes using NBER TAXSIM
are described in Section 4. The quantitative findings along with the
sensitivity analysis, evidence on welfare gains distribution, and the results
of the counterfactual reforms are reported in Section 5. The linearization
bias is discussed in Section 6. Section 7 concludes.

##  2 Model

Economic Environment. To study the welfare effects of the changes in labor
income taxes, I build a static model of married couples along the lines of
Kaygusuz (2010) and Bick and Fuchs-Schündeln (2017b).222 Here and thereafter,
I use “labor income taxes” and “taxes” interchangeably. Consider an economy
populated by $N$ married couples. In each couple, spouses choose joint private
consumption, $c$, males choose how much to work (intensive margin), $h^{m}$,
and females choose whether to work (extensive margin) and, conditional on
participation, how much to work (intensive margin), $h^{f}$. I abstract from
modeling the extensive margin for males because in the data their
participation rates are traditionally high and demonstrate little variation
over time. Hence the households in the model are either single-earner or dual-
earner couples. To model the extensive margin of labor supply for women, I
assume that each couple draw fixed utility cost of work $q_{i}$ from a
distribution $F_{i}\left(q_{i}\right)$. This cost is incurred when a wife
enters the labor market. I interpret it as a utility loss that is related to
inconvenience of scheduling joint work for both spouses or childcare
responsibilities (Cho and Rogerson, 1988). Modeling the extensive margin with
the fixed cost of work allows generating the distribution of hours that is
consistent with the data. In particular, as Figure A.1 reports, the empirical
distribution of married women’s annual working hours has a little mass at low
number of hours.

The wages of a male and a female in couple $i$ are denoted by $w^{m}_{i}$ and
$w^{f}_{i}$ correspondingly. The tax and transfer system is introduced by
function $T\left(w^{m}_{i}h^{m},w^{f}_{i}h^{f},\theta\right)$. It embodies all
labor income taxes and transfers and may feature non-separabilities between
the arguments. I assume that $T(\cdot)$ is piecewise linear, so that the
spouses face locally constant marginal tax rates. The policy reform is modeled
in a flexible way by allowing $T(\cdot)$ to be a function of a treatment
parameter $\theta$. Changes in $\theta$ capture any arbitrary tax reform. In
what follows, I focus on small reforms ($d\theta\approx 0$). Furthermore, I
assume that there no other externalities than those operating through the
government budget.

Household Optimization. The utility maximization problem of couple $i$ is
given by

$\max_{c,h^{m},h^{f}}U_{i}\left(c,h^{m},h^{f}\right)=v_{i}\left(c,h^{m},h^{f}\right)-q_{i}\cdot\mathbbm{1}\\{h^{f}>0\\}$
(1)
$\text{s.t.}~{}~{}~{}~{}~{}c=w^{m}_{i}h^{m}+w^{f}_{i}h^{f}-T\left(w^{m}_{i}h^{m},w^{f}_{i}h^{f},\theta\right)$
(2)

where $v_{i}(\cdot)$ is a well-behaved utility function, and
$\mathbbm{1}\\{h^{f}>0\\}$ takes the value one if the wife works and zero
otherwise. Similarly to Eissa et al. (2008), my formulation accounts for the
presence of income effects. This problem can be solved in two stages. First,
conditional on wife’s participation, the couple choose the hours of work.
Second, the wife makes a participation decision at the optimal level of
working hours. Consider these stages in turn.

First, given that both spouses work, $h^{m}>0$ and $h^{f}>0$, the optimal
allocation for couple $i$ is characterized by the following first-order
conditions:

$(1-\tau^{j}_{i}(\theta))w^{j}_{i}\frac{\partial
v_{i}\left(c^{2}_{i},h^{m,2}_{i},h^{f}_{i}\right)}{\partial c}=-\frac{\partial
v_{i}\left(c^{2}_{i},h^{m,2}_{i},h^{f}_{i}\right)}{\partial
h^{j}},~{}~{}~{}~{}~{}~{}~{}j=m,f$ (3)

where $c^{2}_{i}$ denotes the optimal consumption in a dual-earner couple,
$h^{m,2}_{i}$ and $h^{f}_{i}$ denote the optimal male’s and female’s hours of
work in a dual-earner couple, and the effective marginal tax rates on male’s
and female’s earnings are denoted by $\tau^{j}_{i}(\theta)\equiv\partial
T\left(w^{m}_{i}h^{m}_{i},w^{f}_{i}h^{f}_{i},\theta\right)/\partial\left(w^{j}_{i}h^{j}_{i}\right)$,
with $j=m,f$. Note that these marginal tax rates include the marginal claw-
back on special provisions of the tax code such as deductions or tax credits.
To simplify notation, I omit explicit dependence of marginal tax rates on
$\theta$.

Next, at the second stage, the wife makes a decision whether to enter the
labor market or not. There exists a threshold level of fixed cost of work,
$\bar{q}_{i}$, such that the wife chooses to work if the utility of the dual-
earner couple is greater than or equal to the utility of the single-earner
couple. This threshold is given by

$\bar{q}_{i}=v_{i}\left(c^{2}_{i},h^{m,2}_{i},h^{f}_{i}\right)-v_{i}\left(c^{1}_{i},h^{m,1}_{i},0\right)$
(4)

where $c^{1}_{i}\equiv
w^{m}_{i}h^{m,1}_{i}-T\left(w^{m}_{i}h^{m,1}_{i},0,\theta\right)$ denotes the
optimal consumption of the single-earner couple. Consumption allocations of
the dual-earner and single-earner couples are connected through the following
equation:

$c^{2}_{i}=w^{m}_{i}h^{m,2}_{i}+w^{f}_{i}h^{f}_{i}-T\left(w^{m}_{i}h^{m,2}_{i},w^{f}_{i}h^{f}_{i},\theta\right)=c^{1}_{i}+(1-a_{i}(\theta))\left[w^{m}_{i}\left(h^{m,2}_{i}-h^{m,1}_{i}\right)+w^{f}_{i}h^{f}_{i}\right]$
(5)

where
$a_{i}(\theta)\equiv\left[T\left(w^{m}_{i}h^{m,2}_{i},w^{f}_{i}h^{f}_{i},\theta\right)-T\left(w^{m}_{i}h^{m,1}_{i},0,\theta\right)\right]/\left(w^{m}_{i}\left(h^{m,2}_{i}-h^{m,1}_{i}\right)+w^{f}_{i}h^{f}_{i}\right)$
is a participation tax rate of the couple (Prescott, 2004). It captures the
change in tax liability as a share of the change in earnings following the
wife’s decision to participate. In the quantitative part of this paper, I
assume that the husband’s earnings do not change if the wife enters the labor
market, and hence $a_{i}$ is effectively the women’s participation tax rate.
To simplify notation, I omit explicit dependence of participation tax rates on
$\theta$.

To evaluate the welfare effects of a tax policy reform, I obtain the
compensated (Hicksian) consumption and labor supply by solving the expenditure
minimization problem. This dual problem is given by

$\min_{c,h^{m},h^{f}}c-w^{m}_{i}h^{m}-w^{f}_{i}h^{f}+T\left(w^{m}_{i}h^{m},w^{f}_{i}h^{f},\theta\right)$
(6)
$\text{s.t.}~{}~{}~{}~{}~{}v_{i}\left(c,h^{m},h^{f}\right)-q_{i}\cdot\mathbbm{1}\\{h^{f}>0\\}\geq\bar{U}_{i}$
(7)

where $\bar{U}_{i}$ is some fixed level of utility.

Similarly to problem (1)-(2), I solve it in two stages. First, given that both
spouses work, $h^{m}>0$ and $h^{f}>0$, the solution is characterized by the
following first-order conditions:

$(1-\tau^{j}_{i})w^{j}_{i}\frac{\partial
v_{i}\left(\tilde{c}^{2}_{i},\tilde{h}^{m,2}_{i},\tilde{h}^{f}_{i}\right)}{\partial
c}=-\frac{\partial
v_{i}\left(\tilde{c}^{2}_{i},\tilde{h}^{m,2}_{i},\tilde{h}^{f}_{i}\right)}{\partial
h^{j}},~{}~{}~{}~{}~{}~{}~{}j=m,f$ (8)
$v_{i}\left(\tilde{c}^{2}_{i},\tilde{h}^{m,2}_{i},\tilde{h}^{f}_{i}\right)=\bar{U}_{i}+q_{i}$
(9)

From (8)-(9), I get compensated consumption,
$\tilde{c}^{2}_{i}=\tilde{c}^{2}_{i}\left(\cdot\right)$, and labor supply,
$\tilde{h}^{m,2}_{i}=\tilde{h}^{m,2}_{i}\left(\cdot\right)$ and
$\tilde{h}^{f}_{i}=\tilde{h}^{f}_{i}\left(\cdot\right)$, for dual-earner
couples. Using these compensated functions, I write down the expenditure
function for a dual-earner couple:

$E^{2}_{i}\left(\bar{U}_{i}+q_{i},\theta\right)=\tilde{c}^{2}_{i}(\cdot)-w^{m}_{i}\tilde{h}^{m,2}_{i}(\cdot)-w^{f}_{i}\tilde{h}^{f}_{i}(\cdot)+T\left(w^{m}_{i}\tilde{h}^{m,2}_{i}(\cdot),w^{f}_{i}\tilde{h}^{f}_{i}(\cdot),\theta\right)$
(10)

This expenditure function is evaluated at the current (pre-reform) tax and
transfer system and depends on the treatment parameter $\theta$ both directly
and through compensated consumption and labor supply.

For a single-earner couple, the solution to the dual problem is characterized
by

$(1-\tau^{m}_{i})w^{m}_{i}\frac{\partial
v_{i}\left(\tilde{c}^{1}_{i},\tilde{h}^{m,1}_{i},0\right)}{\partial
c}=-\frac{\partial
v_{i}\left(\tilde{c}^{1}_{i},\tilde{h}^{m,1}_{i},0\right)}{\partial h^{m}}$
(11) $v_{i}\left(\tilde{c}^{1}_{i},\tilde{h}^{m,1}_{i},0\right)=\bar{U}_{i}$
(12)

and it delivers compensated consumption
$\tilde{c}^{1}_{i}=\tilde{c}^{1}_{i}\left(\cdot\right)$ and male’s labor
supply, $\tilde{h}^{m,1}_{i}=\tilde{h}^{m,1}_{i}\left(\cdot\right)$.

The expenditure function of a single-earner couple is given by

$E^{1}_{i}\left(\bar{U}_{i},\theta\right)=\tilde{c}^{1}_{i}(\cdot)-w^{m}_{i}\tilde{h}^{m,1}_{i}(\cdot)+T\left(w^{m}_{i}\tilde{h}^{m,1}_{i}(\cdot),0,\theta\right)$
(13)

Given utility $\bar{U}_{i}$, the wife chooses to enter the labor market if
$E^{2}_{i}\left(\bar{U}_{i}+q_{i},\theta\right)\leq
E^{1}_{i}\left(\bar{U}_{i},\theta\right)$, and not participate otherwise.
Therefore, I write the expenditure function in the following way:

$E_{i}\left(\bar{U}_{i},q_{i},\theta\right)=\min\Big{\\{}E^{1}_{i}\left(\bar{U}_{i},\theta\right),E^{2}_{i}\left(\bar{U}_{i}+q_{i},\theta\right)\Big{\\}}$
(14)

Defining a compensated threshold of the fixed cost of work, $\tilde{q}_{i}$,
such that
$E^{2}_{i}\left(\bar{U}_{i}+\tilde{q}_{i},\theta\right)=E^{1}_{i}\left(\bar{U}_{i},\theta\right)$,
and plugging (10) and (13) into this definition, I obtain the equation
connecting compensated consumption in dual-earner and single-earner couples:

$\tilde{c}^{2}_{i}=\tilde{c}^{1}_{i}+(1-a_{i})\left[w^{m}_{i}\left(\tilde{h}^{m,2}_{i}-\tilde{h}^{m,1}_{i}\right)+w^{f}_{i}\tilde{h}^{f}_{i}\right]$
(15)

Furthermore, evaluating (9) and (12) at $\tilde{q}_{i}$, I get

$\tilde{q}_{i}=v_{i}\left(\tilde{c}^{2}_{i},\tilde{h}^{m,2}_{i},\tilde{h}^{f}_{i}\right)-v_{i}\left(\tilde{c}^{1}_{i},\tilde{h}^{m,1}_{i},0\right)$
(16)

Setting $\bar{U}_{i}$ to be equal to the indirect utility obtained from
problem (1)-(2), I guarantee that the solution to the expenditure minimization
problem is consistent with the solution to the utility maximization problem.

Aggregate Labor Supply. Since the focus of the paper is in aggregate welfare
effects of tax reforms, it is natural to ask how does a tax policy affect the
aggregate labor supply of couples. Turning from individual optimization to the
aggregates, I write down aggregate compensated labor supply:

$\tilde{L}=\sum_{i=1}^{N}\Bigg{[}\underbrace{\int_{0}^{\tilde{q}_{i}}\tilde{h}^{m,2}_{i}\left((1-\tau^{m}_{i})w^{m}_{i},(1-\tau^{f}_{i})w^{f}_{i},\bar{U}_{i}+q_{i}\right)dF_{i}\left(q_{i}\right)}_{\text{compensated
labor supply of males in dual-earner couples}}+\\\
\underbrace{\int_{0}^{\tilde{q}_{i}}\tilde{h}^{f}_{i}\left((1-\tau^{m}_{i})w^{m}_{i},(1-\tau^{f}_{i})w^{f}_{i},\bar{U}_{i}+q_{i}\right)dF_{i}\left(q_{i}\right)}_{\text{compensated
labor supply of females in dual-earner couples}}+\\\
\underbrace{\int_{\tilde{q}_{i}}^{\infty}\tilde{h}^{m,1}_{i}\left((1-\tau^{m}_{i})w^{m}_{i},0,\bar{U}_{i}\right)dF_{i}\left(q_{i}\right)}_{\text{compensated
labor supply of males in single-earner couples}}\Bigg{]}$ (17)

The assumption about separability of $q_{i}$ implies that the utility net of
fixed cost of work, i.e. $v_{i}=\bar{U}_{i}+q_{i}$, is independent of the
realization of $q_{i}$. Therefore, I rewrite (17) as

$\tilde{L}=\sum_{i=1}^{N}\Bigg{[}\underbrace{F_{i}\left(\tilde{q}_{i}\right)}_{\text{affected
by
$a_{i}(\theta)$}}\underbrace{\left(\tilde{h}^{m,2}_{i}+\tilde{h}^{f}_{i}\right)}_{\text{affected
by $\tau^{m}_{i}(\theta)$ and
$\tau^{f}_{i}(\theta)$}}+\underbrace{\left(1-F_{i}\left(\tilde{q}_{i}\right)\right)}_{\text{affected
by $a_{i}(\theta)$}}\underbrace{\tilde{h}^{m,1}_{i}}_{\text{affected by
$\tau^{m}_{i}(\theta)$}}\Bigg{]}$ (18)

where $F_{i}\left(\tilde{q}_{i}\right)$ denotes the probability of being a
dual-earner couple or, alternatively, the individual probability of the
woman’s participation. This term can be also interpreted as the married
women’s participation rate. Aggregate compensated labor supply depends on
extensive and intensive margins of spousal labor supply. The former affects
$\tilde{L}$ through the woman’s participation rate $F_{i}(\tilde{q}_{i})$ that
is driven by the participation tax rate $a_{i}$. The latter affects
$\tilde{L}$ through the working hours that are driven by the marginal tax
rates $\tau^{m}_{i}$ and $\tau^{f}_{i}$. These labor supply behavioral
responses are the key factors in assessing the efficiency gains of tax policy
reforms.

Elasticities. To evaluate the welfare effects of a tax reform, I reformulate
my results in terms of the compensated elasticities of labor supply. First,
define the compensated participation elasticity as the percentage change in
the woman’s individual participation rate resulting from a one percentage
change in the participation net-of-tax rate:

$\eta_{i}\equiv\frac{\partial
F_{i}\left(\tilde{q}_{i}\right)}{\partial\left(1-a_{i}\right)}\cdot\frac{1-a_{i}}{F_{i}\left(\tilde{q}_{i}\right)}$
(19)

Next, define the compensated elasticity of working hours for males and females
as the percentage change in working hours resulting from a one percentage
change in the effective marginal net-of-tax rate:

$\varepsilon^{m,\iota}_{i}\equiv\frac{\partial\tilde{h}^{m,\iota}_{i}}{\partial\left(1-\tau^{m}_{i}\right)}\cdot\frac{1-\tau^{m}_{i}}{\tilde{h}^{m,\iota}_{i}},~{}~{}~{}~{}~{}~{}~{}\iota=1,2$
(20)
$\varepsilon^{f}_{i}\equiv\frac{\partial\tilde{h}^{f}_{i}}{\partial\left(1-\tau^{f}_{i}\right)}\cdot\frac{1-\tau^{f}_{i}}{\tilde{h}^{f}_{i}}$
(21)

Finally, for dual-earner couples, define the cross-elasticities of working
hours as the percentage change in individual’s working hours resulting from a
one percentage change in the effective marginal net-of-tax rate of his/her
spouse. These elasticities are absent in the framework with singles.

$\varepsilon^{mf}_{i}\equiv\frac{\partial\tilde{h}^{m,2}_{i}}{\partial\left(1-\tau^{f}_{i}\right)}\cdot\frac{1-\tau^{f}_{i}}{\tilde{h}^{m,2}_{i}}$
(22)
$\varepsilon^{fm}_{i}\equiv\frac{\partial\tilde{h}^{f}_{i}}{\partial\left(1-\tau^{m}_{i}\right)}\cdot\frac{1-\tau^{m}_{i}}{\tilde{h}^{f}_{i}}$
(23)

Efficiency Loss. To study the effects of a tax reform on individual welfare, I
define the measure of excess burden using the equivalent variation. Under this
definition, the excess burden from the current tax and transfer system
$\theta$ is the difference between the sum of money that the couple is willing
to pay to move to the economy without distortionary taxes and transfers and
collected tax revenue (Auerbach, 1985):

$D_{i}\left(\bar{U}_{i},q_{i},\theta\right)=E_{i}\left(\bar{U}_{i},q_{i},\theta\right)-E_{i}\left(\bar{U}_{i},q_{i},0\right)-R\left(\bar{U}_{i},q_{i},\theta\right)$
(24)

where $R\left(\bar{U}_{i},q_{i},\theta\right)$ is given by

$R\left(\bar{U}_{i},q_{i},\theta\right)=\begin{cases}T\left(w^{m}_{i}\tilde{h}^{m,2}_{i}(\cdot),w^{f}_{i}\tilde{h}^{f}_{i}(\cdot),\theta\right),~{}~{}~{}~{}~{}\text{if}~{}~{}q_{i}<\tilde{q}_{i}\\\
T\left(w^{m}_{i}\tilde{h}^{m,1}_{i}(\cdot),0,\theta\right),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{otherwise}\end{cases}$
(25)

Aggregate excess burden under a tax and transfer system $\theta$ is defined as
the sum of excess burdens over all couples:

$D=\sum_{i=1}^{N}\int_{0}^{\infty}D_{i}\left(\bar{U}_{i},q_{i},\theta\right)dF_{i}\left(q_{i}\right)$
(26)

Aggregate efficiency loss $D$ measures additional revenue that can be
collected, keeping couples at their initial utility levels $\bar{U}_{i}$, if
the tax and transfer system $\theta$ were to be replaced by a lump-sum tax
system. With heterogeneous agents, aggregate excess burden depends on the
initial income distribution except under very strong conditions on preferences
(Auerbach, 1985; Auerbach and Hines, 2002). In Section 5.4, I discuss the
sensitivity of my results to alternative initial income distributions.

Plugging (14) and (25) into (24), I rewrite aggregate excess burden (26) as

$D=\sum_{i=1}^{N}\Bigg{[}\int_{0}^{\tilde{q}_{i}}\left(E^{2}_{i}\left(\bar{U}_{i}+q_{i},\theta\right)-T\left(w^{m}_{i}\tilde{h}^{m,2}_{i}(\cdot),w^{f}_{i}\tilde{h}^{f}_{i}(\cdot),\theta\right)\right)dF_{i}\left(q_{i}\right)+\\\
\int_{\tilde{q}_{i}}^{\infty}\left(E^{1}_{i}\left(\bar{U}_{i},\theta\right)-T\left(w^{m}_{i}\tilde{h}^{m,1}_{i}(\cdot),0,\theta\right)\right)dF_{i}\left(q_{i}\right)-\int_{0}^{\infty}E_{i}\left(\bar{U}_{i},q_{i},0\right)dF_{i}\left(q_{i}\right)\Bigg{]}$
(27)

I focus on a small tax reform ($d\theta\approx 0$), and to capture its welfare
effects, I study how aggregate efficiency loss $D$ changes with $\theta$. At
this step, I refer to the envelope theorem and the assumption that there are
no other externalities beyond those operating through the government budget,
and show that any arbitrary small reform affects the expenditure function only
through mechanical revenue effect, i.e.
$dE^{2}_{i}\left(\bar{U}_{i}+q_{i},\theta\right)/d\theta=\partial
T\left(w^{m}_{i}\tilde{h}^{m,2}_{i}(\cdot),w^{f}_{i}\tilde{h}^{f}_{i}(\cdot),\theta\right)/\partial\theta$
for dual-earner couples and
$dE^{1}_{i}\left(\bar{U}_{i},\theta\right)/d\theta=\partial
T\left(w^{m}_{i}\tilde{h}^{m,1}_{i}(\cdot),0,\theta\right)/\partial\theta$ for
single-earner couples. Since the spouses optimize and there are no non-tax or
non-transfer externalities, a small tax reform does not have the first-order
effects on the expenditure functions and utility. In turn, the first-order
effects come from externalities that operate through the government budget. In
particular, when the spouses adjust their working hours or labor force
participation, they create fiscal externality on all the other households.
Having stated this, it follows from definition (24) that the effect of any
arbitrary small tax reform on economic efficiency is captured by the
behavioral revenue effect (“fiscal externality”) or the difference between
mechanical revenue effect, $\partial T_{i}/\partial\theta$, and total revenue
effect, $dT_{i}/d\theta$.

Differentiating (27) and using the result from the previous paragraph, I
obtain

$\frac{dD}{d\theta}=-\sum_{i=1}^{N}\Bigg{[}\tau^{m}_{i}w^{m}_{i}\frac{\partial\tilde{h}^{m,2}_{i}}{\partial\theta}F_{i}\left(\tilde{q}_{i}\right)+\tau^{m}_{i}w^{m}_{i}\frac{\partial\tilde{h}^{m,1}_{i}}{\partial\theta}\left(1-F_{i}\left(\tilde{q}_{i}\right)\right)+\\\
\tau^{f}_{i}w^{f}_{i}\frac{\partial\tilde{h}^{f}_{i}}{\partial\theta}F_{i}\left(\tilde{q}_{i}\right)+a_{i}\left[w^{m}_{i}\left(\tilde{h}^{m,2}_{i}-\tilde{h}^{m,1}_{i}\right)+w^{f}_{i}\tilde{h}^{f}_{i}\right]\frac{\partial
F_{i}\left(\tilde{q}_{i}\right)}{\partial\theta}\Bigg{]}$ (28)

The effect of a small tax reform on economic efficiency is driven by
behavioral responses along intensive and extensive margins of labor supply.
The first three terms in (28) stand for reform-induced changes in the working
hours. The last term captures the effect of reform-induced changes in female
labor force participation.

Denote aggregate labor income by

$W\equiv\sum_{i=1}^{N}\left(w^{m}_{i}\tilde{h}^{m,2}_{i}+w^{f}_{i}\tilde{h}^{f}_{i}\right)F_{i}\left(\tilde{q}_{i}\right)+w^{m}_{i}\tilde{h}^{m,1}_{i}\left(1-F_{i}\left(\tilde{q}_{i}\right)\right)$
(29)

so that the expected labor income shares are given by $s^{m,2}_{i}\equiv
w^{m}_{i}\tilde{h}^{m,2}_{i}F_{i}\left(\tilde{q}_{i}\right)/W$ for males in
dual-earner couples, $s^{f}_{i}\equiv
w^{f}_{i}\tilde{h}^{f}_{i}F_{i}\left(\tilde{q}_{i}\right)/W$ for females, and
$s^{m,1}_{i}\equiv
w^{m}_{i}\tilde{h}^{m,1}_{i}\left(1-F_{i}\left(\tilde{q}_{i}\right)\right)/W$
for males in single-earner couples. Finally, in Proposition 1, I state the
main formula that expresses the reform-induced change in economic efficiency
in terms of the empirically estimable objects.

Proposition 1 (Reform-Induced Change in Economic Efficiency). The effect of
any arbitrary small tax reform $d\theta\approx 0$ on economic efficiency,
captured by marginal excess burden as a fraction of aggregate labor income, is
given by

$\frac{dD/d\theta}{W}=\sum_{i=1}^{N}\Bigg{[}\left(\frac{\tau^{m}_{i}}{1-\tau^{m}_{i}}\cdot\frac{d\tau^{m}_{i}}{d\theta}\varepsilon^{m,2}_{i}+\frac{\tau^{m}_{i}}{1-\tau^{f}_{i}}\cdot\frac{d\tau^{f}_{i}}{d\theta}\varepsilon^{mf}_{i}\right)s^{m,2}_{i}+\frac{\tau^{m}_{i}}{1-\tau^{m}_{i}}\cdot\frac{d\tau^{m}_{i}}{d\theta}\varepsilon^{m,1}_{i}s^{m,1}_{i}+\\\
\left(\frac{\tau^{f}_{i}}{1-\tau^{f}_{i}}\cdot\frac{d\tau^{f}_{i}}{d\theta}\varepsilon^{f}_{i}+\frac{\tau^{f}_{i}}{1-\tau^{m}_{i}}\cdot\frac{d\tau^{m}_{i}}{d\theta}\varepsilon^{fm}_{i}\right)s^{f}_{i}+\frac{a_{i}}{1-a_{i}}\cdot\frac{da_{i}}{d\theta}\eta_{i}\left(s^{m,2}_{i}+s^{f}_{i}-\frac{F_{i}\left(\tilde{q}_{i}\right)}{1-F_{i}\left(\tilde{q}_{i}\right)}s^{m,1}_{i}\right)\Bigg{]}$
(30)

Proof. See Appendix.

One of the advantages of using the sufficient statistics approach is the
transparency of the results. In particular, using equation (30), I can
decompose the aggregate effect of a tax reform into behavioral effects that
operate through the men’s working hours (the first and the third terms), the
women’s working hours (the fourth term), the spousal cross-effects of working
hours (the second and the fifth terms), and, finally, the women’s
participation margin (the last term). It is useful to emphasize the difference
between equation (30) and one that is obtained when households are modeled as
single individuals. The first difference comes from the cross-elasticities,
$\varepsilon^{mf}_{i}$ and $\varepsilon^{fm}_{i}$, that capture the changes in
individual’s working hours induced by the changes in spousal net-of-tax rate.
The existing estimates of these elasticities are different from zero (Blau and
Kahn, 2007), even though the empirical evidence is quite limited. In Section
5, I show that these terms matter for the overall effect. Second, my framework
also accounts for the changes in husband’s working hours following the wife’s
decision to join the labor force. Neither of these terms are present in the
setting without couples.

In what follows, I construct the effective marginal and participation tax
rates, reform-induced changes in the effective tax rates, and the expected
labor income shares using the Current Population Survey data and the Internet
NBER TAXSIM tax calculator. Furthermore, I take the estimates of the
elasticities from the literature and, to study the sensitivity of the results,
consider different ranges of values. Next, using these empirical estimates in
equation (30), I quantify the changes in economic efficiency caused by the tax
reforms implemented in the United States over the last four decades.

##  3 Data

I use the data from the Annual Social and Economic Supplement (ASEC) of the
Current Population Survey (CPS), or the “March CPS”.333 The CPS data is
extracted from IPUMS at https://cps.ipums.org/cps. See Flood et al. (2020).
The March CPS contains data on annual earnings corresponding to the previous
year. I define earnings as the sum of wage income and self-employment income.
To be consistent with the model from Section 2, I focus on different-sex
married couples with working husbands in which both a husband and a wife are
aged 25-54. Since I do not model education or retirement decisions, I do not
include younger or older individuals. Furthermore, I exclude the couples where
husbands do not have a reasonably strong labor market attachment. In
particular, I drop the households where a husband earns less than a time-
varying minimum threshold defined as one-half of the federal minimum wage
times 520 hours (13 weeks at 40 hours per week) which amounts to annual
earnings of $1885 in 2012 USD (Guvenen et al., 2014). Wives in my sample
either work or not. To be consistent with the idea of reasonable labor market
attachment of workers, I drop the couples where wives work but have annual
earnings less than the time-varying minimum threshold described above. Using
an alternative threshold of $100 in 2012 USD does not change the results.

It is well known that the household survey data is subject to several caveats.
First, there are two types of non-response: non-interview and item non-
response. In the March CPS, non-response is accounted for by imputing missing
values. As pointed out by Meyer et al. (2015), the non-response rates in the
major U.S. household surveys, including CPS, are growing over time. While
imputation may cause serious problems for studying the trends over time, it
works well if the object of interest is the cross-sectional distribution of
individuals.444 For this reason, it is quite common in the literature,
studying the trends, to drop the observations with imputed data (Ziliak et
al., 2011). I consider each tax reform separately, and use the cross-sectional
distribution of couples in each pre-reform year. By this reason, I do not
exclude the observations with imputed values. Second, earnings in the survey
data are subject to bottom- or top-coding. Until 1995, the CPS used the
traditional top-coding when the respondents, who reported income over the
maximum allowed value, were assigned this maximum value. I drop all the top-
coded and bottom-coded observations. In 1996-2010, the CPS used a replacement
value system. The main difference with the traditional top-coding is that
incomes above the maximum threshold are replaced by mean income of the other
high-income individuals with similar demographic characteristics. Since 2011,
the CPS has been using the rank proximity swapping procedure that preserves
the distribution of values above the threshold. I do not drop the observations
that are imputed using these procedures.

The summary statistics for the pre-reform years—1986, 1992, 2000, and 2017—is
shown in Tables A.1 and A.2. Several things worth emphasizing. First, mean and
median annual hours of males have barely changed since the 1980s. In turn,
mean and median annual hours of females have significantly increased. For
example, the median went from 1400 hours in 1986 to 1872 hours in 2017, a 34
percent increase. Second, the employment rates among married women do not
display such a significant increase. This observation echoes the discussion
about stagnating female labor force participation in the United States (Blau
and Kahn, 2013). Third, in the 2010s the U.S. women are ahead of men in
college education, even though in 1986 they were significantly behind (Goldin,
2014). Finally, although not reported, the share of families receiving welfare
benefits, such as the Aid to Families with Dependent Children (AFDC) and
Temporary Assistance for Needy Families (TANF) or the Supplemental Nutrition
Assistance Program (SNAP), is small in my sample. Therefore, in the
simulations, I do not account for welfare benefits that are lost when a wife
enters the labor market.

##  4 The U.S. Tax Reforms

###  4.1 Background

My goal is to evaluate the welfare gains of the labor income tax changes on
married couples induced by four reforms in the United States: the Tax Reform
Act of 1986, the Omnibus Budget Reconciliation Act of 1993, the Economic
Growth and Tax Relief Reconciliation Act of 2001, and the Tax Cuts and Jobs
Act of 2017. While they affected various parts of the tax code, I focus
exclusively on labor income taxes. In what follows, I describe the main
reform-induced changes in the tax schedule for married couples filing jointly.

The top-left panel of Figure A.2 shows that the Tax Reform Act of 1986
significantly decreased the number of tax brackets. Despite the marginal tax
rates were reduced for almost all the range of taxable income, they went up at
the bottom of the income distribution and for the interval between $60000 and
$69000 in 2012 USD. The top tax rate was decreased from 50 to 38.5 percent for
tax year 1987, and then down to 28 percent in tax year 1988.555 In 1988-1990,
the marginal tax rate structure included a 5 percent surtax within some range
of taxable income. Next, as reported in Tables LABEL:tab:eitc_parameters and
OA.2, there was an expansion in the EITC, standard deductions, and personal
exemptions.

Next, the top-right panel of Figure A.2 reports that the Omnibus Budget
Reconciliation Act of 1993 increased the top tax rate from 31 to 39.6 percent.
Following the reform, the couples faced higher marginal tax rates for the
taxable income above $222000 in 2012 USD. However, on the other hand, the OBRA
1993 significantly expanded the EITC, thus benefiting low-income households
(Kleven, 2020). As a result, this reform could potentially have different
effects on married women with working husbands than on the other groups
sensitive to the changes in the tax and transfer system, such as single women
(Eissa and Liebman, 1996; Eissa et al., 2008). The reason is that, in general,
they belong to different parts of the income distribution, and the former are
more likely affected by higher marginal tax rates rather than the EITC
expansion.

The Economic Growth and Tax Relief Reconciliation Act of 2001 resulted into
lower marginal tax rates for most tax brackets including the top income tax
rate that went down from 39.6 to 35 percent. Moreover, the standard deduction
for married couples filing jointly was increased relative to single filers. In
particular, in 2000, the standard deduction for a married couple filing
jointly was 67 percent higher than for a single filer, while in 2003 it became
100 percent higher.

Finally, the Tax Cuts and Jobs Act of 2017 featured changes in the federal
income tax brackets and reduction in the marginal tax rates over almost the
whole range of taxable income. The top income tax rate was decreased from 39.6
to 37 percent. Next, the standard deductions were increased, while personal
exemptions were eliminated and itemized deductions were reduced. The
individual income tax changes under the TCJA 2017 are effective for tax years
2018-2025. Moreover, there was a permanent shift from the Consumer Price Index
(CPI) to the U.S. Chained Consumer Price Index (C-CPI-U) for indexing the tax
brackets over time.

###  4.2 Reform-Induced Changes in Tax Rates

I calculate tax liabilities and reform-induced changes in tax rates for each
couple in my sample using NBER TAXSIM calculator.666 See Feenberg and Coutts
(1993) for introduction to TAXSIM. Further details are available at
https://www.nber.org/taxsim/. This software provides accurate representation
of the U.S. tax code and allows capturing the heterogeneous effects of tax
reforms on households. In related papers, Bick and Fuchs-Schündeln (2017b) and
Bick et al. (2019) emphasize the importance of accounting for nonlinearities
of the labor income tax code for studying the effects of tax and transfer
system on labor supply of married couples.

For each spouse in my sample, TAXSIM returns the federal, state, and the
Federal Income Contributions Act (FICA) tax liabilities as well as
corresponding effective marginal tax rates. In Online Appendix, I provide the
full list of input variables and describe how I fill each field. To be
consistent with the model from Section 2, I abstract from all non-labor
income. Next, because I do not explicitly model the children and childcare
expenses, I set the number of children to two for each couple (this a median
value for all the years in Tables A.1 and A.2). I also assume that all couples
choose joint filing.777 Married taxpayers in the United States pretty rarely
file separate returns. According to IRS Income Statistics, in 2017 tax year
about 95 percent of married couples filed jointly. Finally, I assume that all
the couples live in Michigan, a “typical area” in terms of state income
taxation. Thus, heterogeneity in tax liabilities is solely driven by
heterogeneity in couples’ earnings. When I allow for variation in the factors
that remain fixed in my analysis, the results do not significantly change.

To construct the participation and marginal tax rates, I need to know the
potential earnings of all spouses, including non-working women. If a woman
works, then I set her potential earnings to be equal to the actual earnings.
If a woman does not work, her potential earnings are equal to her income in
the case of entering the labor market. Since they are not observable, I apply
a two-stage Heckman procedure to impute these earnings. I use the exclusion
restrictions that the husband’s earnings and the number of children under 6 do
not directly influence the woman’s earnings (Mulligan and Rubinstein, 2008;
Bick and Fuchs-Schündeln, 2017b). Next, to obtain the expected labor income
shares, I use the predicted probability of labor force participation as an
empirical analogue of $F_{i}\left(\tilde{q}_{i}\right)$. Finally, I assume
that workers bear the full incidence of employer payroll taxes. In this case,
the proper measure of pre-tax labor income is equal to earnings plus the
employer’s share (50%) of the FICA tax. Hence, when I construct the tax rates,
I divide all of them by the factor of $(1+0.5\cdot\text{FICA})$.

For each woman in the sample, I construct an effective participation tax rate.
In particular, for a woman in couple $i$ it is given by

$a_{it}=\frac{T_{t}\left(y^{m}_{it},\hat{y}^{f}_{it},Dem_{it}\right)-T_{t}\left(y^{m}_{it},0,Dem_{it}\right)}{\hat{y}^{f}_{it}}$
(31)

where $y^{m}_{it}$ denotes the husband’s taxable income in year $t$,
$\hat{y}^{f}_{it}$ denotes the wive’s taxable income in year $t$, $Dem_{it}$
denotes other TAXSIM inputs. I assume that the husband’s earnings do not
change when the wife enters the labor market.

The effective marginal tax rate in TAXSIM is calculated as the additional tax
liabilities resulting from changing the taxable income by 10 cents. For
example, for women it is given by

$\tau^{f}_{it}=\frac{T_{t}\left(y^{m}_{it},\hat{y}^{f}_{it}+\$0.1,Dem_{it}\right)-T_{t}\left(y^{m}_{it},\hat{y}^{f}_{it},Dem_{it}\right)}{\$0.1}$
(32)

The left panel of Figure 2 reports the income-weighted mean effective marginal
and participation tax rates for my sample. The tax rates include federal,
state, and the FICA tax rates. Grey shaded areas represent the periods of
reforms when the changes in taxes came into effect. On the one hand, the TRA
1986, the EGTRRA 2001, and the TCJA 2017 resulted in a decrease in the mean
effective tax rates for the married couples. On the contrary, the OBRA 1993
led to an increase in the effective tax rates.888 This is different from its
effect on single women. In particular, their effective marginal and
participation tax rates dropped, mainly due to the EITC expansion (Eissa et
al., 2008) The drop in the tax rates resulting from the Tax Relief,
Unemployment Insurance Reauthorization, and Job Creation Act of 2010, that I
do not analyze, was driven by a temporary reduction in the FICA tax.

Figure 2: Left panel — Mean effective marginal and female participation tax
rates. Right panel — Reform-induced changes in the mean effective marginal and
female participation tax rates

Left Panel Notes: Marginal (solid blue) and participation (dashed red) tax
rates include federal, state, and the FICA tax rates. Marginal tax rate series
represents the mean marginal tax rate for males and females. Shaded areas
indicate reform years. Right Panel Notes: To construct the changes, I apply
the post-reform federal tax rules to the pre-reform taxable income and impute
the post-reform federal income tax liabilities. The bars show the mean changes
in the effective tax rates induced by the federal tax reforms. Changes in the
marginal tax rates are calculated jointly for males and females.

The time series in the left panel of Figure 2 are driven by various factors,
such as macroeconomic effects and behavioral responses. Furthermore, they
capture the joint changes in the federal, state, and FICA tax rates. To
isolate the changes in the effective tax rates separately induced by each
federal tax reforms, I use the following procedure. For each spouse in pre-
reform year $t$, I apply the federal tax rules of post-reform year $t+x$ to
their year-$t$ real taxable income, keeping the state and FICA tax rules at
year-$t$ level. Then I use the actual pre-reform and imputed post-reform
federal tax liabilities to calculate the changes in the effective marginal and
women’s participation tax rates solely driven by the federal tax reforms. In
this way, I find the empirical analogues of $d\tau^{j}_{i}/d\theta$ and
$da_{i}/d\theta$. The mean changes in the tax rates are reported in the right
panel of Figure 2.

##  5 Quantitative Results

###  5.1 Baseline Parameterization

To quantify the welfare effects of each tax reform separately, I take the pre-
reform effective marginal and participation tax rates (31) and (32), the
reform-induced changes in tax rates, the expected labor income shares, and the
estimates of elasticities from the literature, and plug them into expression
(30). Because I assume that husband’s earnings do not change when his wife
starts working, the last bracket in (30) simplifies to $s^{f}_{i}$. The
welfare gains are defined as (30) taken with the negative sign.

The rich existing empirical evidence suggests that women respond more along
the participation margin rather than working hours margin when face tax and
transfer changes (Bargain et al., 2014). As for men’s elasticities, many
studies suggest that their elasticity of working hours is very low and can
almost be ignored for welfare purposes (Meghir and Phillips, 2010). The
evidence on the cross-wage labor supply effects, that is crucial for my
framework, is quite limited. In a related work, Gayle and Shephard (2019),
using the tax schedule corresponding to 2006, obtain the model-simulated
working hours cross-elasticity is -0.08 for married men and -0.17 for married
women. These numbers are, in general, consistent with existing empirical
estimates (Blau and Kahn, 2007). Among other factors, the hours cross-
elasticities may depend on the number of young children (Blundell et al.,
2018). It is also worth noting that women’s labor supply elasticities feature
significant heterogeneity at the micro level, and any aggregate elasticity
depends on the particular economic environment (Attanasio et al., 2018).

Under the baseline parameterization, I set the elasticity of male hours to
0.05, the elasticity of female hours to 0.1, women’s participation elasticity
to 0.6, the hours cross-elasticity of males to -0.05, and the hours cross-
elasticity of females to -0.1. In Section 5.2, I conduct sensitivity analysis
by varying the magnitudes of elasticities.

Beyond evaluating the actual welfare effects of tax reforms, it is also
instructive to consider a benchmark case corresponding to a representative
couple. This environment features no heterogeneity in income, tax rates, and
tax rate changes. I assume $\tau^{m}=\tau^{f}$ because of tax system
jointness. The pre-reform tax rates, $\tau$ and $a$, are given by the mean
effective marginal and participation tax rates reported in the left panel of
Figure 2. The reform-induced tax changes, $d\tau/d\theta$ and $da/d\theta$,
are given by the mean changes in the tax rates reported in the right panel of
Figure 2. Overall, the expression for the effect of a small tax reform on
economic efficiency is simplified to

$\frac{dD/d\theta}{W}=\frac{\tau}{1-\tau}\cdot\frac{d\tau}{d\theta}\left[\left(\varepsilon^{m}+\varepsilon^{mf}\right)s^{m}+\left(\varepsilon^{f}+\varepsilon^{fm}\right)s^{f}\right]+\frac{a}{1-a}\cdot\frac{da}{d\theta}\eta
s^{f}$ (33)

where $s^{m}=w^{m}h^{m}/W$ denotes the labor income share of men, and
$s^{f}=w^{f}h^{f}F\left(\tilde{q}\right)/W$ denotes the labor income share of
women.

Table 1 reports the welfare gains for married couples resulting from each of
four considered U.S. tax reforms. Total welfare gains (column 6) are
decomposed into four parts: behavioral effects created along the intensive
margin of men’s labor supply (column 1), the intensive margin of women’s labor
supply (column 2), the participation margin of women (column 3), and, finally,
the spousal cross-effects of working hours (column 4). To emphasize the
quantitative importance of cross-effects, in column (5), I show total welfare
gains if we abstract from them. Effectively, column (5) is the sum of own
effects given in columns (1), (2), and (3). Column (7) displays the welfare
gains calculated in a representative-couple economy, according to (33).
Furthermore, to facilitate comparison across the reforms, in column (9), I
report the welfare gain per dollar spent.

Table 1: Welfare effects of labor income tax changes on married couples with
working husbands

| Welfare gain, % of aggregate labor income |  |
---|---|---|---
Reform |  | Intensive
---
Males
| Intensive
---
Females
| Extensive
---
Females
| Cross-
---
Effects
| Total
---
w/o C.E.
Total | RC |  | Tax Liab.
---
Reduc., %
|  $\Delta$ Welfare/
---
$ Spent
| (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
TRA86 | 0.19 | 0.18 | 0.45 | -0.27 | 0.82 | 0.55 | 0.44 | 7.20 | 1.08
OBRA93 | -0.01 | -0.02 | -0.15 | 0.03 | -0.18 | -0.16 | -0.16 | 0.27 | 0.63
EGTRRA01 | 0.09 | 0.12 | 0.40 | -0.17 | 0.61 | 0.44 | 0.42 | 7.19 | 1.07
TCJA17 | 0.10 | 0.17 | 0.57 | -0.22 | 0.84 | 0.62 | 0.58 | 6.58 | 1.10

Notes: Welfare gains are calculated using (30) taken with the negative sign. I
set $\varepsilon^{m}=0.05$, $\varepsilon^{f}=0.15$, $\varepsilon^{mf}=-0.05$,
$\varepsilon^{fm}=-0.1$, and $\eta=0.6$. The pre-reform tax rates and reform-
induces changes in tax rates are calculated using NBER TAXSIM applied to the
ASEC CPS data. Column (5) shows total welfare gains when the cross-effects are
ignored, and calculated as $(1)+(2)+(3)$. Column (6) shows total welfare
gains, and calculated as $(4)+(5)$. Column (7) shows the welfare gains in a
representative-couple economy. Column (9) is calculated as $(8)/[(8)-(6)]$,
where (8) is the decrease in tax liabilities as a share of labor income before
behavioral responses.

First, Table 1 shows that reform-induced changes in federal income tax rates
result in the welfare gains that range from -0.16 to 0.62 percent of aggregate
labor income. These numbers reflect the welfare effects that are driven by the
labor supply behavioral responses. Three reforms—the TRA 1986, the EGTRRA
2001, and the TCJA 2017—created aggregate welfare gains, while the OBRA 1993
created welfare loss. Second, it follows from comparing columns (5) and (6)
that the spousal working hours cross-effects are quantitatively important and
therefore should not be ignored in the welfare analysis of policies.
Otherwise, it may lead to overestimation of the welfare effects. For example,
if I abstract from the cross-effects, I overestimate the welfare gains from
the TCJA 2017 by 34.6%. While this number seems to be high, the sensitivity
analysis in Section 5.2 confirms the argument that the spousal cross-effects
remain quantitatively important under any reasonable values of elasticities.
The next conclusion from Table 1 is that the women’s participation margin
accounts for the bulk of total welfare gains. Again, ignoring this factor may
lead to sizable bias in the estimates of policy welfare effects (Kleven and
Kreiner, 2006; Eissa et al., 2008). Another lesson from Table 1 is that a
representative couple model, that uses the income-weighted mean tax rates and
mean changes in tax rates, deliver the results that are close to ones reported
in column (6). Hence, if we are primarily interested in assessing aggregate
welfare gains from tax reforms, a representative agent model may be a
reasonable candidate device for this purpose. Finally, from the values in
column (9), I conclude that the welfare gains vary between 0.63 and 1.10 USD
per dollar spent.

The results reported in Table 1 provide a transparent decomposition of the
aggregate welfare effects, thus highlighting one of the advantages of a
sufficient statistics approach. Furthermore, when I construct the reform-
induced changes in tax rates and then use them in formula (30), I focus solely
on the tax-driven labor supply behavioral responses. My results are not
affected by the other incentives created by the reforms. For example, the TRA
1986 reform led to a shift of income that was previously labeled as corporate
income to personal income (Guvenen and Kaplan, 2017). Despite these clear
advantages, I also discuss the potential caveats and the ways to address them.
First, my framework assumes that the reforms are small, and the measured
efficiency gains represent a first-order approximation of the true effects. I
use the pre-reform tax rates in (30), however the reforms change the tax
rates. The first-order approximation overstates the welfare gains of tax
reductions and understates the welfare losses of tax increases (Kleven, 2021).
The only case that can be considered as a large reform is the reduction in the
top tax rate during the TRA 1986. However, when I use the trapezoid
approximation to evaluate the effects of this reform, the results do not
dramatically change. Furthermore, since I consider each reform separately, I
take into account the second-order effects across the reforms. Second, the
elasticities may move in response to the reforms as well. Blau and Kahn (2007)
and Heim (2007) report that between the 1970s and 2000s there was a dramatic
reduction in own- and cross-elasticities of married women’s labor supply. To
address this caveat, I conduct sensitivity analysis using different
combinations of elasticities. For example, moving from the 1980s to the 2010s
may be viewed as moving from the “high-elasticity” to the “low-elasticity”
parameterization described in Section 5.2. Furthermore, there is substantial
heterogeneity in labor supply elasticities, and the aggregate elasticities are
not structural parameters (Attanasio et al., 2018). To address this concern, I
conduct sensitivity analysis and construct the lower and upper bounds for the
welfare effects using reasonable ranges of elasticities. Finally, I derive
(30) under the assumption that the tax and transfer function is linear. In
Section 6, I show that the linearity assumption leads to overestimation of the
welfare gains and characterize this linearization bias.

Table 2: Welfare effects of labor income tax changes on married couples,
sensitivity analysis

| Welfare gain, % of aggregate labor income |  |
---|---|---|---
Reform |  | Intensive
---
Males
| Intensive
---
Females
| Extensive
---
Females
| Cross-
---
Effects
| Total
---
w/o C.E.
Total | RC |  | Tax Liab.
---
Reduc., %
|  $\Delta$ Welfare/
---
$ Spent
| (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
“Upper-Bound” Parameterization: $\varepsilon^{m}=0.1$, $\varepsilon^{f}=0.2$,
$\varepsilon^{mf}=0$, $\varepsilon^{fm}=-0.05$, $\eta=0.8$
TRA86 | 0.39 | 0.24 | 0.60 | -0.08 | 1.23 | 1.15 | 1.03 | 7.20 | 1.19
OBRA93∗ | 0.00 | -0.01 | -0.10 | 0.04 | -0.12 | -0.07 | -0.25 | 0.27 | 0.79
EGTRRA01 | 0.18 | 0.16 | 0.54 | -0.04 | 0.88 | 0.84 | 0.77 | 7.19 | 1.13
TCJA17 | 0.19 | 0.23 | 0.76 | -0.06 | 1.18 | 1.12 | 1.03 | 6.58 | 1.21
“Lower-Bound” Parameterization: $\varepsilon^{m}=0$, $\varepsilon^{f}=0.1$,
$\varepsilon^{mf}=-0.1$, $\varepsilon^{fm}=-0.15$, $\eta=0.4$
TRA86 | 0.00 | 0.12 | 0.30 | -0.47 | 0.42 | -0.05 | -0.14 | 7.20 | 0.99
OBRA93∗ | -0.02 | -0.03 | -0.20 | 0.01 | -0.25 | -0.25 | -0.07 | 0.27 | 0.53
EGTRRA01 | 0.00 | 0.08 | 0.27 | -0.30 | 0.35 | 0.05 | 0.06 | 7.19 | 1.01
TCJA17 | 0.00 | 0.12 | 0.38 | -0.37 | 0.49 | 0.12 | 0.13 | 6.58 | 1.02
“High-Elasticity” Parameterization: $\varepsilon^{m}=0.1$,
$\varepsilon^{f}=0.2$, $\varepsilon^{mf}=-0.1$, $\varepsilon^{fm}=-0.15$,
$\eta=0.8$
TRA86 | 0.39 | 0.24 | 0.60 | -0.47 | 1.23 | 0.75 | 0.57 | 7.20 | 1.12
OBRA93 | -0.02 | -0.03 | -0.20 | 0.04 | -0.25 | -0.21 | -0.22 | 0.27 | 0.57
EGTRRA01 | 0.18 | 0.16 | 0.54 | -0.30 | 0.88 | 0.57 | 0.54 | 7.19 | 1.09
TCJA17 | 0.19 | 0.23 | 0.76 | -0.37 | 1.18 | 0.81 | 0.76 | 6.58 | 1.14
“Low-Elasticity” Parameterization: $\varepsilon^{m}=0$, $\varepsilon^{f}=0.1$,
$\varepsilon^{mf}=0$, $\varepsilon^{fm}=-0.05$, $\eta=0.4$
TRA86 | 0.00 | 0.12 | 0.30 | -0.08 | 0.42 | 0.34 | 0.32 | 7.20 | 1.05
OBRA93 | 0.00 | -0.01 | -0.10 | 0.01 | -0.12 | -0.11 | -0.11 | 0.27 | 0.72
EGTRRA01 | 0.00 | 0.08 | 0.27 | -0.04 | 0.35 | 0.31 | 0.29 | 7.19 | 1.05
TCJA17 | 0.00 | 0.12 | 0.38 | -0.06 | 0.49 | 0.44 | 0.40 | 6.58 | 1.07
Baseline Parameterization + Participation Elasticity Varies by Income Quintile
TRA86 | 0.19 | 0.18 | 0.23 | -0.27 | 0.61 | 0.33 | - | 7.21 | 1.05
OBRA93 | -0.01 | -0.02 | -0.21 | 0.03 | -0.24 | -0.21 | - | 0.27 | 0.56
EGTRRA01 | 0.09 | 0.12 | 0.28 | -0.17 | 0.49 | 0.32 | - | 7.19 | 1.05
TCJA17 | 0.10 | 0.17 | 0.34 | -0.22 | 0.61 | 0.39 | - | 6.58 | 1.06

Notes: Welfare gains are calculated using (30) taken with the negative sign.
The pre-reform tax rates and reform-induces changes in tax rates are
calculated using NBER TAXSIM applied to the ASEC CPS data. Column (5) shows
total welfare gains when the cross-effects are ignored, and calculated as
$(1)+(2)+(3)$. Column (6) shows total welfare gains, and calculated as
$(4)+(5)$. Column (7) shows the welfare gains in a representative-couple
economy. Column (9) is calculated as $(8)/[(8)-(6)]$, where (8) is the
decrease in tax liabilities as a share of labor income before behavioral
responses. In the last panel, the participation elasticity takes values
1/0.8/0.6/0.4/0.2 for the bottom/second/third/fourth/top couple’s income
quintiles, keeping the mean participation elasticity equal to 0.6.
∗ Since the OBRA 1993 increased or left unchanged the tax rates for most
spouses in the sample, I use the parameters from the “lower-bound” scenario in
the panel corresponding to the “upper-bound” scenario and vice-versa.

###  5.2 Sensitivity Analysis

To explore the sensitivity of my results, I consider several alternative
parameterizations of elasticities. The results are reported in Table 2. First,
I begin with the “upper-bound” scenario. Under this parameterization, own
elasticities have reasonably high values ($\varepsilon^{m}=0.1$,
$\varepsilon^{f}=0.2$, and $\eta=0.8$) and cross-elasticities have reasonably
low values ($\varepsilon^{mf}=0$ and $\varepsilon^{fm}=-0.05$) for the TRA
1986, the EGTRRA 2001, and TCJA 2017 reforms, i.e. the reforms that feature
reductions in the mean effective tax rates. On the contrary, for the OBRA
1993, I assume low own elasticities ($\varepsilon^{m}=0$,
$\varepsilon^{f}=0.1$, and $\eta=0.4$) and high cross-elasticities
($\varepsilon^{mf}=-0.1$ and $\varepsilon^{fm}=-0.15$). Under the “upper-
bound” parameterization, the welfare gains range from -0.07 to 1.15 percent of
aggregate labor income. Next, I consider the opposite scenario, namely, the
“lower-bound” parameterization of elasticities. I flipped the values and
assume low own elasticities and high cross-elasticities for the TRA 1986, the
EGTRRA 2001, and TCJA 2017 reforms, and vice versa for the OBRA 1993. In this
case, the welfare gains range from -0.25 to 0.12 percent of aggregate labor
income. Overall, the first two panels of Table 2 can inform us about the
bounds on efficiency gains resulting from the U.S. tax reforms.

Next, I consider two parameterizations labeled as “high-elasticity” and “low-
elasticity”. In particular, I set all the elasticities to high values in the
former case, and low values in the latter case. The third and fourth panels of
Table 2 may facilitate the comparison of reforms that were conducted in
different time periods. If married women’s elasticities shrank between the
1970s and 2000s (Blau and Kahn, 2007; Heim, 2007), then the conclusion from
Table 1 that the TCJA 2017 reform created the largest efficiency gains among
four reforms may be reconsidered.

Finally, in the bottom panel of Table 2, I report the results of an exercise
where I allow the participation elasticity to decline in household income. In
particular, I assign values 1/0.8/0.6/0.4/0.2 to the
first/second/third/fourth/fifth income quintiles, keeping the mean
participation elasticity equal to 0.6. Under this parameterization, total
efficiency gains range from -0.21 to 0.39 percent of aggregate labor income.
These numbers are lower than in the baseline scenario because high-income
couples have smaller participation elasticities and hence benefit less from
tax reductions. In turn, low-income couples benefit more, but the aggregate
measure of efficiency gains masks heterogeneity in welfare effects. Beyond all
these findings, the results from Table 2 reinforce my claim that cross-
elasticities and participation elasticities quantitatively important and
should be accounted for in the welfare analysis of economic policies.

(a) TRA 1986 reform

(b) OBRA 1993 reform

(c) EGTRRA 2001 reform

(d) TCJA 2017 reform

Figure 3: Distribution of reform-induced welfare gains among couples

Notes: Welfare gains are calculated under a baseline parameterization of
elasticities.

###  5.3 Welfare Gains Distribution

So far I consider the aggregate welfare effects of tax reforms, however the
use of microdata combined with TAXSIM allows studying the distribution of
welfare gains and losses. Indeed, the aggregate effects may mask significant
heterogeneity across households.999 Using the U.S. data, Zidar (2019) finds
that the positive relationship between tax cuts and employment growth is
largely driven by tax cuts for low-income group, rather than high-income
individuals. In this section, I use the baseline parameterization of
elasticities and answer the following questions. How are the efficiency gains
from tax reforms distributed in the population of married couples?
Furthermore, according to Table 1, three reforms created aggregate welfare
gains. Are there any losers? Finally, how do the welfare gains vary by income?

Table 3: Distribution of welfare gains for couples, % of couple’s labor income

Reform | P10 | P25 | P50 | P75 | P90
---|---|---|---|---|---
TRA86 | -0.21 | 0.13 | 0.37 | 0.61 | 0.94
OBRA93 | -1.09 | -0.40 | 0.00 | 0.00 | 0.00
EGTRRA01 | 0.00 | 0.06 | 0.36 | 0.76 | 0.95
TCJA17 | 0.10 | 0.23 | 0.42 | 0.74 | 1.02

Notes: Welfare gains are calculated under a baseline parameterization of
elasticities, and are measured as a share of couple’s labor income.

Figure 3 documents the distribution of welfare gains among couples for each
reform. A simple visual inspection confirms the argument that income tax
changes create heterogeneous welfare effects, and there are both winners and
losers from each reform. In Table 3, I report the percentiles of efficiency
gains distribution. For the TRA 1986, the median welfare gain is equal to 0.37
percent of couple’s labor income. While a substantial fraction of couples win
from the reform, those at the 10th percentile experience welfare loss equal to
0.21 percent of the labor income. On the contrary, couples at the 90th
percentile receive welfare gain of 0.94 percent of labor income. Next,
following the OBRA 1993, the median couple stay welfare neutral. However,
couples at the 10th percentile experience welfare loss of 1.09 percent of
labor income. In the case of the other two reforms, the EGTRRA 2001 and the
TCJA 2017, the values corresponding to the 10th percentile are non-negative.
One more observation that follows from Figure 3 and Table 3 is that the
dispersion of the efficiency gains significantly differs across the reforms.
Apart from the OBRA 1993, where most of couples are welfare neutral, the
P75-P25 ratio for the TRA 1986, the EGTRRA 2001, and the TCJA 2017 is equal to
4.7, 12.7, and 3.2 correspondingly.

Next, in Table 4, I report the fractions of winners, losers, and welfare-
neutral couples. I define winners as those with welfare gains above 0.1
percent of couple’s labor income. Losers are defined as those with welfare
losses greater than 0.1 percent of labor income. Finally, welfare-neutral
couples are those whose absolute values of welfare gains or losses do not
exceed 0.1 percent of labor income. It follows that, despite the TRA 1986
created aggregate welfare gains, it left 12.3 percent of couples with welfare
losses. In turn, while the OBRA 1993 created aggregate welfare losses, for
about two-thirds of the married couples this reform was welfare-neutral. In
the case of EGTRRA 2001 and the TCJA 2017 the share of winners is equal to
69.6 and 90.3 percent correspondingly.

Table 4: Fractions of winners, losers, and welfare-neutral couples

Reform | Winners, % | Losers, % | Neutral, %
---|---|---|---
TRA86 | 78.7 | 12.3 | 9.1
OBRA93 | 1.4 | 31.2 | 67.4
EGTRRA01 | 69.6 | 0.3 | 30.1
TCJA17 | 90.3 | 0.6 | 9.0

Notes: Welfare gains are calculated under a baseline parameterization of
elasticities. Winners are defined as couples with welfare gains above 0.1
percent of couple’s labor income. Losers are defined as couples with welfare
losses greater than 0.1 percent of labor income. Welfare-neutral couples are
defined as those whose absolute values of welfare gains or losses do not
exceed 0.1 percent of labor income.

(a) TRA 1986 reform

(b) OBRA 1993 reform

(c) EGTRRA 2001 reform

(d) TCJA 2017 reform

Figure 4: Welfare gains and income distribution, couples

Notes: Welfare gains are measured as a percentage of the labor income. Each
dot represents 5 percent of the sample. The grey shaded area represents the
interval between “lower-bound” and “upper-bound” elasticity parameterizations.

Finally, in Figure 4, I explore how the welfare gains vary by income. Each dot
represents 5 percent of the sample, and the grey shaded areas represent the
interval between “lower-bound” and “upper-bound” elasticity parameterizations.
Figure 4 clearly shows that efficiency gains change nonlinearly with income.
There are two general patterns. First, the TRA 1986, the OBRA 1993 (excluding
the bottom 10 percent), and the TCJA 2017 can be characterized as monotonic
tax reforms (Bierbrauer et al., 2021) and they resulted in monotonic
relationships between welfare gains and income. Overall, richer taxpayers
benefited from these reforms more than poorer taxpayers. Second, the OBRA 1993
and the EGTRRA 2001 reforms demonstrate a U-shaped pattern in the welfare
gains. In this case, the main winners of the reforms are located at the lower
and upper ends of the income distribution. Interestingly, Hotchkiss et al.
(2012) and Hotchkiss et al. (2021) discover similar patterns of the welfare
gains despite using very different methodology. Overall, the results from this
section reassure that despite a representative couple model can be a
reasonable candidate for assessing aggregate efficiency gains, it does not
capture rich heterogeneity and misses important distributional aspects of tax
reforms.

###  5.4 Counterfactual Tax Reforms

In this section, I conduct two sets of counterfactual tax reforms aimed at
addressing the following questions. First, how does the pre-reform income
distribution matter for my results? Second, how do the initial conditions—pre-
reform income distribution and tax law—jointly matter for the estimates of
welfare gains?

In Table 5, I report the results for the first set of counterfactual reforms.
In this exercise, I take the couples’ income distribution in pre-reform year
$t$ (for example, in 1986), and apply the pre- and post-reform $X$’s (for
example, the TCJA 2017) tax laws. Table 5 consists of four panels, where each
panel represents the income distribution that I use. For example, Panel A
shows the results for four reforms applied to the 1986 income distribution.
The first column displays the reforms. Column (9) reports the percentage
difference between counterfactual and actual welfare gains per dollar spent,
shown in column (8). By construction, the results in the first line of Panel A
coincide with one from Table 1, and hence there is zero in column (9). The
results from the bottom line of Panel A should be interpreted in the following
way: if the TCJA 2017 were to be applied to the 1986 income distribution,
total welfare gains would be 0.40 percent of aggregate labor income. It
follows from column (9) that this counterfactual 2017 TCJA reform would
deliver 2.68% less welfare gain per dollar spent than the actual TCJA 2017
(reported in the bottom line of Panel D). In turn, the first line of Panel D
shows that if the TRA 1986 reform were to be applied to the 2017 income
distribution, then the welfare gain per dollar spent would be 5.48% higher
than from the actual TRA 1986. Overall, the results from Table 5 suggest that
despite the initial income distribution matters, it has limited quantitative
importance. Counterfactual welfare gains per dollar spent do not differ by
more than 7.54% from the actual ones.

Table 5: Welfare effects of tax reforms applied to counterfactual income
distribution

| Welfare gain, % of aggregate labor income |  |
---|---|---|---
Reform |  | Intensive
---
Males
| Intensive
---
Females
| Extensive
---
Females
| Cross-
---
Effects
Total | RC |  | Tax Liab.
---
Reduc., %
|  $\Delta$ Welfare/
---
$ Spent
Diff., %
| (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
Panel A: Tax Reforms Applied to Pre-TRA86 Distribution of Couples
TRA86 | 0.19 | 0.18 | 0.45 | -0.27 | 0.55 | 0.44 | 7.20 | 1.08 | 0.00
OBRA93 | -0.01 | -0.02 | -0.13 | 0.02 | -0.14 | -0.14 | 0.29 | 0.68 | +7.54
EGTRRA01 | 0.09 | 0.11 | 0.36 | -0.16 | 0.40 | 0.37 | 7.46 | 1.06 | -0.80
TCJA17 | 0.09 | 0.12 | 0.36 | -0.18 | 0.40 | 0.37 | 5.76 | 1.07 | -2.68
Panel B: Tax Reforms Applied to Pre-OBRA93 Distribution of Couples
TRA86 | 0.19 | 0.22 | 0.53 | -0.30 | 0.63 | 0.51 | 7.38 | 1.09 | +1.09
OBRA93 | -0.01 | -0.02 | -0.15 | 0.03 | -0.16 | -0.16 | 0.27 | 0.63 | 0.00
EGTRRA01 | 0.08 | 0.12 | 0.39 | -0.16 | 0.43 | 0.40 | 7.38 | 1.06 | -0.32
TCJA17 | 0.09 | 0.14 | 0.41 | -0.18 | 0.45 | 0.42 | 5.87 | 1.08 | -1.88
Panel C: Tax Reforms Applied to Pre-EGTRRA01 Distribution of Couples
TRA86 | 0.33 | 0.31 | 0.82 | -0.48 | 0.97 | 0.76 | 10.23 | 1.11 | +2.11
OBRA93 | -0.04 | -0.04 | -0.18 | 0.07 | -0.19 | -0.20 | -0.97 |  |
EGTRRA01 | 0.09 | 0.12 | 0.40 | -0.17 | 0.44 | 0.42 | 7.19 | 1.07 | 0.00
TCJA17 | 0.10 | 0.14 | 0.44 | -0.20 | 0.48 | 0.45 | 6.19 | 1.08 | -1.80
Panel D: Tax Reforms Applied to Pre-TCJA17 Distribution of Couples
TRA86 | 0.29 | 0.42 | 1.13 | -0.52 | 1.32 | 1.05 | 10.62 | 1.14 | +5.48
OBRA93 | -0.03 | -0.05 | -0.22 | 0.07 | -0.24 | -0.25 | -0.96 |  |
EGTRRA01 | 0.08 | 0.13 | 0.48 | -0.17 | 0.52 | 0.49 | 7.15 | 1.08 | +1.18
TCJA17 | 0.10 | 0.17 | 0.57 | -0.22 | 0.62 | 0.58 | 6.58 | 1.10 | 0.00

Notes: Welfare gains are calculated using (30) taken with the negative sign
and under a baseline parameterization of elasticities. In each exercise, I
take the distribution of couples corresponding to some pre-reform year, as
indicated in four panels, and calculate the welfare effects from applying the
reform that is shown in the left-most column. Column (5) shows total welfare
gains, and calculated as $(1)+(2)+(3)+(4)$. Column (6) shows the welfare gains
in a representative-couple economy. Column (8) is calculated as
$(7)/[(7)-(5)]$, where (7) is the decrease in tax liabilities as a share of
labor income before behavioral responses. Column (9) shows the percentage
difference relative to the actual welfare gains from the reform.

Next, I conduct a set of counterfactual reforms, where I take the distribution
of couples and the tax law corresponding to some pre-reform year $t$ (for
example, in 1986) and apply the post-reform $X$’s (for example, the TCJA 2017)
tax law. In other words, in this example, I study the welfare consequences of
moving from the pre-TRA 1986 economy to the post-TCJA 2017 economy. Table 6
reports the results. I do not conduct the backward reforms between two
consecutive reforms (e.g., I do not consider the welfare consequences of
moving from the pre-OBRA 1993 economy to the post-TRA 1986 economy) because
the welfare effects are likely to be negligible in these cases. Panel A
suggests two interesting findings. First, moving from the pre-TRA 1986 economy
to the post-EGTRRA 2001 and post-TCJA 2017 economies leads to substantial
reduction in tax liabilities, 17.85 and 22.28 percent, relative to the actual
TRA 1986 reform (7.20 percent), and higher efficiency gains, 0.88 and 1.19
percent of aggregate labor income (the actual TRA 1986 results in 0.55 percent
of aggregate labor income). However, when I make the efficiency gains
comparable, column (8) shows that the actual TRA 1986 generated more welfare
gain per dollar spent than the alternative considered counterfactual reforms.

Table 6: Welfare effects of counterfactual tax reforms

| Welfare gain, % of aggregate labor income |
---|---|---
Reform |  | Intensive
---
Males
| Intensive
---
Females
| Extensive
---
Females
| Cross-
---
Effects
Total | RC |  | Tax Liab.
---
Reduc., %
|  $\Delta$ Welfare/
---
$ Spent
| (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8)
Panel A: Tax Reforms Applied to Pre-TRA86 Distribution of Couples and Tax Law
TRA86 | 0.19 | 0.18 | 0.45 | -0.27 | 0.55 | 0.44 | 7.20 | 1.08
OBRA93 | 0.19 | 0.17 | 0.35 | -0.27 | 0.44 | 0.29 | 7.73 | 1.06
EGTRRA01 | 0.27 | 0.27 | 0.75 | -0.41 | 0.88 | 0.74 | 17.85 | 1.05
TCJA17 | 0.36 | 0.38 | 1.02 | -0.58 | 1.19 | 0.96 | 22.28 | 1.06
Panel B: Tax Reforms Applied to Pre-OBRA93 Distribution of Couples and Tax Law
TRA86 | — | — | — | — | — | — | — | —
OBRA93 | -0.01 | -0.02 | -0.15 | 0.03 | -0.16 | -0.16 | 0.27 | 0.63
EGTRRA01 | 0.06 | 0.09 | 0.26 | -0.12 | 0.29 | 0.27 | 10.09 | 1.03
TCJA17 | 0.13 | 0.19 | 0.51 | -0.25 | 0.57 | 0.52 | 14.69 | 1.04
Panel C: Tax Reforms Applied to Pre-EGTRRA01 Distribution of Couples and Tax
Law
TRA86 | 0.09 | 0.08 | 0.25 | -0.15 | 0.27 | 0.22 | -0.74 |
OBRA93 | — | — | — | — | — | — | — | —
EGTRRA01 | 0.09 | 0.12 | 0.40 | -0.17 | 0.44 | 0.42 | 7.19 | 1.07
TCJA17 | 0.15 | 0.23 | 0.69 | -0.31 | 0.76 | 0.70 | 12.16 | 1.07
Panel D: Tax Reforms Applied to Pre-TCJA17 Distribution of Couples and Tax Law
TRA86 | 0.03 | 0.02 | 0.05 | -0.05 | 0.05 | -0.02 | -6.40 |
OBRA93 | -0.03 | -0.06 | -0.26 | 0.07 | -0.27 | -0.29 | -7.38 |
EGTRRA01 | — | — | — | — | — | — | — | —
TCJA17 | 0.10 | 0.17 | 0.57 | -0.22 | 0.62 | 0.58 | 6.58 | 1.10

Notes: Welfare gains are calculated using (30) taken with the negative sign
and under a baseline parameterization of elasticities. In each exercise, I
take the distribution of couples and the tax law corresponding to some pre-
reform year, as indicated in four panels, and calculate the welfare effects
from applying the reform that is shown in the left-most column. Column (5)
shows total welfare gains, and calculated as $(1)+(2)+(3)+(4)$. Column (6)
shows the welfare gains in a representative-couple economy. Column (8) is
calculated as $(7)/[(7)-(5)]$, where (7) is the decrease in tax liabilities as
a share of labor income before behavioral responses.

##  6 Efficiency Loss and Nonlinear Taxation of Couples

###  6.1 Linearization Bias

The framework in Section 2 is elaborated under linear tax and transfer
function. Despite the real tax and transfer schedules often feature
nonlinearities, this assumption is widely used in the literature studying the
efficiency gains of tax reforms (Chetty, 2009). How sizable is the bias in the
estimates of welfare gains resulting from the linearity assumption? In this
section, I address this question by extending the framework of Blomquist and
Simula (2019), who study linearization bias through the lens of the model with
singles, to the setting with couples.

I use a version of the model from Section 2. In particular, consider an
economy populated by couples with preferences
$v\left(c,y_{m},y_{f},\upsilon_{m},\upsilon_{f}\right)$ where $y_{m}$ and
$y_{f}$ are taxable incomes of a male and a female, $\upsilon_{m}$ and
$\upsilon_{f}$ are idiosyncratic preference parameters jointly drawn from
continuous distribution $\varGamma$. To focus on the bias that comes from tax
function linearization, I abstract from the labor force participation margin
and consider an economy populated by dual-earner couples, i.e. in all couples
both spouses are employed. Furthermore, I state the problem in terms of
taxable rather than labor income.101010 Feldstein (1999) suggests that the
relevant statistic for calculating efficiency loss is the elasticity of
taxable income because people can adjust their behavior along different
margins, not only labor supply. The couple’s budget constraint is given by
$c=y_{m}+y_{f}-T\left(y_{m},y_{f},\theta\right)+I$ where $I$ is lump-sum non-
taxable income. Following the similar steps as in Section 2, I obtain the
expression for marginal deadweight loss, $dD/d\theta$.

Now suppose that the original tax and transfer function $T$ is replaced by a
linearized function $T^{L}$ that delivers the same solution as the original
problem, $\left(c^{*},y^{*}_{m},y^{*}_{f}\right)$. In particular, the latter
is described by proportional tax rates $\tau_{m}=\partial
T\left(y_{m},y_{f},\theta\right)/\partial y_{m}$ and $\tau_{f}=\partial
T\left(y_{m},y_{f},\theta\right)/\partial y_{f}$ and a lump-sum component.
Namely,

$T^{L}\left(y_{m},y_{f},\tau_{m},\tau_{f}\right)=\tau_{m}(\theta)y_{m}+\tau_{f}(\theta)y_{f}+T^{*}$
(34)

where I set
$T^{*}=y_{m}^{*}+y_{f}^{*}-c^{*}-\tau_{m}y^{*}_{m}+\tau_{f}y^{*}_{f}+I$.
Again, following the steps from Section 2 and using the linearized budget
constraint (34), I obtain the expression for marginal deadweight loss,
$dD^{L}/d\theta$.

How does the original reform-induced efficiency loss, $dD/d\theta$, differ
from the change in efficiency loss under a linearized tax and transfer
function, $dD^{L}/d\theta$? Proposition 2 characterizes these objects and
reveals that they depend on two sets of terms, utility curvature and tax
function curvature. In particular, from binding
$v\left(c,y_{m},y_{f},\upsilon_{m},\upsilon_{f}\right)=\bar{U}$, obtain the
inverse, $c=\psi\left(y_{m},y_{f},\upsilon_{m},\upsilon_{f},\bar{U}\right)$.
Next, denote
$\psi^{\prime\prime}_{ij}\equiv\partial^{2}\psi(\cdot)/\partial\tilde{y}_{i}\partial\tilde{y}_{j}$,
$T^{\prime\prime}_{ij}\equiv\partial^{2}T(\cdot)/\partial\tilde{y}_{i}\partial\tilde{y}_{j}$,
$T^{\prime\prime}_{i\theta}\equiv\partial^{2}T(\cdot)/\partial\tilde{y}_{i}\partial\theta$,
and, finally, $T^{\prime}_{i}\equiv\partial T(\cdot)/\partial y_{i}$. Then,
$\psi$-terms account for utility curvature and $T$-terms account for tax
function curvature.

Proposition 2 (Efficiency Loss under Nonlinear Taxation of Couples). Under
nonlinear tax function $T$, efficiency loss from any arbitrary small tax
reform $d\theta\approx 0$ is given by

$\frac{dD}{d\theta}=-\int\Bigg{[}\frac{T^{\prime}_{m}\left[\left(\psi^{\prime\prime}_{mf}+T^{\prime\prime}_{mf}\right)T^{\prime\prime}_{f\theta}-\left(\psi^{\prime\prime}_{ff}+T^{\prime\prime}_{ff}\right)T^{\prime\prime}_{m\theta}\right]}{\left(\psi^{\prime\prime}_{mm}+T^{\prime\prime}_{mm}\right)\left(\psi^{\prime\prime}_{ff}+T^{\prime\prime}_{ff}\right)-\left(\psi^{\prime\prime}_{mf}+T^{\prime\prime}_{mf}\right)^{2}}+\\\
\frac{T^{\prime}_{f}\left[\left(\psi^{\prime\prime}_{mf}+T^{\prime\prime}_{mf}\right)T^{\prime\prime}_{m\theta}-\left(\psi^{\prime\prime}_{mm}+T^{\prime\prime}_{mm}\right)T^{\prime\prime}_{f\theta}\right]}{\left(\psi^{\prime\prime}_{mm}+T^{\prime\prime}_{mm}\right)\left(\psi^{\prime\prime}_{ff}+T^{\prime\prime}_{ff}\right)-\left(\psi^{\prime\prime}_{mf}+T^{\prime\prime}_{mf}\right)^{2}}\Bigg{]}d\varGamma\left(\upsilon_{m},\upsilon_{f}\right)$
(35)

Under linearized tax function $T^{L}$, efficiency loss from any arbitrary
small tax reform $d\theta\approx 0$ is given by

$\frac{dD^{L}}{d\theta}=-\int\Bigg{[}\frac{T^{\prime}_{m}\left(\psi^{\prime\prime}_{mf}T^{\prime\prime}_{f\theta}-\psi^{\prime\prime}_{ff}T^{\prime\prime}_{m\theta}\right)}{\psi^{\prime\prime}_{mm}\psi^{\prime\prime}_{ff}-\left(\psi^{\prime\prime}_{mf}\right)^{2}}+\frac{T^{\prime}_{f}\left(\psi^{\prime\prime}_{mf}T^{\prime\prime}_{m\theta}-\psi^{\prime\prime}_{mm}T^{\prime\prime}_{f\theta}\right)}{\psi^{\prime\prime}_{mm}\psi^{\prime\prime}_{ff}-\left(\psi^{\prime\prime}_{mf}\right)^{2}}\Bigg{]}d\varGamma\left(\upsilon_{m},\upsilon_{f}\right)$
(36)

Proof. See Appendix.

To obtain (36) from (35), one should simply set
$T^{\prime\prime}_{mm}=T^{\prime\prime}_{ff}=T^{\prime\prime}_{mf}=0$.
Expressions (35) and (36) are generalized versions of equations (5) and (8)
from Blomquist and Simula (2019) who derive them for the economy populated by
singles. To demonstrate that their result is a special case of Proposition 2,
set to zero all joint terms,
$\psi^{\prime\prime}_{mf}=T^{\prime\prime}_{mf}=0$, in (35), and then marginal
deadweight loss for individual of gender $j$ is given by
$T^{\prime}_{j}T^{\prime\prime}_{j\theta}/\left(\psi^{\prime\prime}_{jj}+T^{\prime\prime}_{jj}\right)$,
and this is (5) in Blomquist and Simula (2019). The linearized version (36) is
given by $T^{\prime}_{j}T^{\prime\prime}_{j\theta}/\psi^{\prime\prime}_{jj}$,
and this is (8) in their paper.

Define the linearization bias as the percentage difference between reform-
induced efficiency loss under linearized and original tax and transfer
functions:

$\Delta=\frac{\frac{dD^{L}}{d\theta}-\frac{dD}{d\theta}}{\frac{dD}{d\theta}}$
(37)

While Proposition 2 suggests that the size of this bias is affected by the
curvatures of utility and tax function, it is instructive to assume the
functional forms for utility $v$ and tax and transfer function $T$ and obtain
the expression for $\Delta$ in terms of model parameters.

###  6.2 Efficiency Loss and HSV Tax Function

I assume that the couple’s preferences are given by

$v\left(c,y_{m},y_{f},\upsilon_{m},\upsilon_{f}\right)=c-\frac{\upsilon_{m}}{\sigma+1}\left(\frac{y_{m}}{\upsilon_{m}}\right)^{\sigma+1}-\frac{\upsilon_{f}}{\sigma+1}\left(\frac{y_{f}}{\upsilon_{f}}\right)^{\sigma+1}$
(38)

where parameter $\sigma$ is the inverse elasticity of taxable income. The
quasilinearity assumption implies that there is no income effect, and hence
the compensated and uncompensated taxable income functions coincide.
Furthermore, $\upsilon_{m}$ and $\upsilon_{f}$ may be interpreted as wages.

Next, I choose the functional form for $T$ to summarize the tax and transfer
system in a simplified way. Heathcote et al. (2017) show that the log-linear
function $T(y)=y-\lambda y^{1-\theta}$ (henceforth, HSV tax function) yields a
remarkably good approximation of the actual tax and transfer system in the
United States. In this specification, parameter $\theta$ is interpreted as a
measure of tax progressivity, and parameter $\lambda$ determines the level of
tax rates. To capture joint and separate taxation of spousal incomes, I
consider the following tax and transfer functions (Bick and Fuchs-Schündeln,
2017b; Borella et al., 2021):

$T\left(y_{m},y_{f}\right)=\lambda\left(y_{m}+y_{f}\right)^{1-\theta}$ (39)
$T\left(y_{m},y_{f}\right)=\tilde{\lambda}y_{m}^{1-\theta}+\tilde{\lambda}y_{f}^{1-\theta}$
(40)

Function (39) describes joint taxation, while function (40) describes separate
taxation. To close the model, I assume that the government uses all tax
revenue to fund the government expenditures, and runs a balanced budget.
Denote by $g$ the share of government consumption in aggregate income.
Finally, I set lump-sum non-taxable income of couples to zero, $I=0$.

I consider a small reform that changes progressivity of the tax and transfer
system, $d\theta\approx 0$. How sizable is the linearization bias? Proposition
3 states that it is given by the ratio between the tax progressivity parameter
$\theta$ and the inverse elasticity of taxable income $\sigma$. Hence, the
magnitude of the bias is jointly determined by a policy parameter (tax
function curvature) and a preference parameter (utility curvature).

Proposition 3 (Linearization Bias with HSV Tax Function). Consider a small
reform that changes tax progressivity, $d\theta\approx 0$. Under both joint
and separate taxation of spousal incomes, the linearization bias is given by

$\Delta=\frac{\theta}{\sigma}$ (41)

Proof. See Appendix.

Higher initial progressivity of the tax system and higher elasticity of
taxable income result in the greater magnitude of the linearization bias or,
alternatively, greater overestimation of aggregate efficiency gain. Using (41)
and estimates of $\theta$ and $\sigma$ from the literature, I can quantify
$\Delta$. Using the information on 1720 estimates of the elasticity of taxable
income from 61 papers, Neisser (2021) report that the majority lies between 0
and 1 with a peak around 0.3 and an excess mass between 0.7 and 1. Next,
Heathcote et al. (2017), using the sample of the U.S. households aged 25-60,
where at least one adult has strong labor market attachment, and the time
period between 2000 and 2006, estimate $\theta=0.181$. Hence, for the range of
taxable income elasticities between 0.2 and 1 (hence, between $\sigma=5$ and
$\sigma=1$), the size of the linearization bias varies between 3.6% (0.181/5
$\times$ 100%) and 18.1% (0.181/1 $\times$ 100%). In other words, aggregate
efficiency gain from a tax reform is overestimated by 3.6-18.1%.

##  7 Conclusion

This paper develops a framework to study the welfare effects of income tax
changes on married couples. I build a static model of couples’ labor supply
that accounts for the presence of both intensive and extensive margins. My
main result is an expression for efficiency gains, resulting from any
arbitrary small tax reform, as a function of (i) labor supply elasticities
capturing the behavioral responses to the tax policy reforms, (ii) pre-reform
tax rates and reform-induced changes in the tax rates, and (iii) labor income
shares. This formula allows to transparently decompose welfare gains into the
effects that operate through the spousal own working hours, spousal cross-
effects of working hours, and the women’s participation margin.

At the next step, I use this expression to quantify the welfare effects of the
labor income tax changes induced by four tax reforms implemented in the United
States: the Tax Reform Act of 1986, the Omnibus Budget Reconciliation Act of
1993, the Economic Growth and Tax Relief Reconciliation Act of 2001, and the
Tax Cuts and Jobs Act of 2017. To parameterize the model, I use the CPS ASEC
data combined with the NBER TAXSIM calculator. My baseline estimates suggest
that these reforms created welfare gains ranging from -0.16 to 0.62 percent of
aggregate labor income. Looking at the forces that shape the efficiency gains,
I find that, first, the bulk of the gains is generated by the women’s labor
force participation responses, and, second, the spousal cross-effects of
working hours are quantitatively important, and abstracting from them leads to
an overestimation of the welfare effects. Although three out of four
considered U.S. tax reforms delivered aggregate welfare gains, I show that
this result masks significant heterogeneity. In fact, each reform created both
winners and losers. Furthermore, the welfare gains are unequally distributed
among the couples with different incomes.

Finally, I show the robustness of my findings to several possible caveats. In
the first set of exercises, I consider alternative parameterizations of
elasticities. It partially allows addressing the concerns about the
sensitivity of the results to the choice of values of elasticities. Next, I
conduct a set of counterfactual reforms aimed at addressing the concerns about
the sensitivity of results to the initial income distribution and the levels
of pre-reform tax rates. Furthermore, I address the concern about the biased
estimates of welfare effects resulting from the assumption about linearity of
the tax and transfer function. To do so, I characterize the linearization bias
under the log-linear tax function that approximates the tax and transfer
system in the United States. Assuming a small reform that translates into a
change in tax progressivity, I show that the linearization bias is given by
the ratio between the tax progressivity parameter and the inverse elasticity
of taxable income. Existing estimates of these objects suggest that the size
of this upward bias lies within the range of 3.6-18.1%.

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## Appendix

### Proof of Proposition 1 (Sufficient Statistics Formula)

This proof extends Eissa et al. (2008) to the framework with couples. First,
differentiate the compensated labor supply functions for dual-earner and
single-earner couples with respect to $\theta$:

$\frac{d\tilde{h}^{m,2}_{i}}{d\theta}=-\sum_{j=m,f}\frac{\partial\tilde{h}^{m,2}_{i}\left((1-\tau^{m}_{i})w^{m}_{i},(1-\tau^{f}_{i})w^{f}_{i},v_{i}\right)}{\partial\left(\left(1-\tau^{j}_{i}\right)w^{j}_{i}\right)}w^{j}_{i}\frac{d\tau^{j}_{i}}{d\theta}=\\\
-\sum_{j=m,f}\left(\frac{\partial\tilde{h}^{m,2}_{i}}{\partial\left(1-\tau^{j}_{i}\right)}\cdot\frac{1-\tau^{j}_{i}}{\tilde{h}^{m,2}_{i}}\right)\cdot\frac{\tilde{h}^{m,2}_{i}}{1-\tau^{j}_{i}}\cdot\frac{d\tau^{j}_{i}}{d\theta}=-\left(\varepsilon^{m,2}_{i}\frac{\tilde{h}^{m,2}_{i}}{1-\tau^{m}_{i}}\cdot\frac{d\tau^{m}_{i}}{d\theta}+\varepsilon^{mf}_{i}\frac{\tilde{h}^{m,2}_{i}}{1-\tau^{f}_{i}}\cdot\frac{d\tau^{f}_{i}}{d\theta}\right)$
(A.1)

and, following similar arguments,

$\frac{d\tilde{h}^{f}_{i}}{d\theta}=-\left(\varepsilon^{f}_{i}\frac{\tilde{h}^{f}_{i}}{1-\tau^{f}_{i}}\cdot\frac{d\tau^{f}_{i}}{d\theta}+\varepsilon^{fm}_{i}\frac{\tilde{h}^{f}_{i}}{1-\tau^{m}_{i}}\cdot\frac{d\tau^{m}_{i}}{d\theta}\right)$
(A.2)
$\frac{d\tilde{h}^{m,1}_{i}}{d\theta}=-\varepsilon^{m,1}_{i}\frac{\tilde{h}^{m,1}_{i}}{1-\tau^{m}_{i}}\cdot\frac{d\tau^{m}_{i}}{d\theta}$
(A.3)

Next, I derive the expression for
$dF_{i}\left(\tilde{q}_{i}\right)/d\theta=\left(\partial
F_{i}\left(\tilde{q}_{i}\right)/\partial\tilde{q}_{i}\right)\cdot\left(d\tilde{q}_{i}/d\theta\right)$.
First, differentiate the expression for threshold $\tilde{q}_{i}$, (16), with
respect to $\theta$:

$\frac{d\tilde{q}_{i}}{d\theta}=\frac{\partial
v_{i}\left(\tilde{c}^{2}_{i},\tilde{h}^{m,2}_{i},\tilde{h}^{f}_{i}\right)}{\partial\tilde{c}^{2}_{i}}\cdot\frac{d\tilde{c}^{2}_{i}}{d\theta}+\frac{\partial
v_{i}\left(\tilde{c}^{2}_{i},\tilde{h}^{m,2}_{i},\tilde{h}^{f}_{i}\right)}{\partial\tilde{h}^{m,2}_{i}}\cdot\frac{d\tilde{h}^{m,2}_{i}}{d\theta}+\frac{\partial
v_{i}\left(\tilde{c}^{2}_{i},\tilde{h}^{m,2}_{i},\tilde{h}^{f}_{i}\right)}{\partial\tilde{h}^{f}_{i}}\cdot\frac{d\tilde{h}^{f}_{i}}{d\theta}-\\\
\frac{\partial
v_{i}\left(\tilde{c}^{1}_{i},\tilde{h}^{m,1}_{i},0\right)}{\partial\tilde{c}^{1}_{i}}\cdot\frac{d\tilde{c}^{1}_{i}}{d\theta}-\frac{\partial
v_{i}\left(\tilde{c}^{1}_{i},\tilde{h}^{m,1}_{i},0\right)}{\partial\tilde{h}^{m,1}_{i}}\cdot\frac{d\tilde{h}^{m,1}_{i}}{d\theta}$
(A.4)

Next, differentiate the equation that connects compensated consumption in
dual-earner and single-earner couples, (15), with respect to $\theta$:

$\frac{d\tilde{c}^{2}_{i}}{d\theta}=\frac{d\tilde{c}^{1}_{i}}{d\theta}+w^{m}_{i}\left(\frac{d\tilde{h}^{m,2}_{i}}{d\theta}-\frac{d\tilde{h}^{m,1}_{i}}{d\theta}\right)+w^{f}_{i}\frac{d\tilde{h}^{f}_{i}}{d\theta}-\Bigg{[}\frac{\partial
T\left(w^{m}_{i}\tilde{h}^{m,2}_{i},w^{f}_{i}\tilde{h}^{f}_{i},\theta\right)}{\partial\left(w^{m}_{i}\tilde{h}^{m,2}_{i}\right)}\cdot\frac{d\tilde{h}^{m,2}_{i}}{d\theta}w^{m}_{i}+\\\
\frac{\partial
T\left(w^{m}_{i}\tilde{h}^{m,2}_{i},w^{f}_{i}\tilde{h}^{f}_{i},\theta\right)}{\partial\left(w^{f}_{i}\tilde{h}^{f}_{i}\right)}\cdot\frac{d\tilde{h}^{f}_{i}}{d\theta}w^{f}_{i}+\frac{dT\left(w^{m}_{i}\tilde{h}^{m,2}_{i},w^{f}_{i}\tilde{h}^{f}_{i},\theta\right)}{d\theta}-\\\
\frac{\partial
T\left(w^{m}_{i}\tilde{h}^{m,1}_{i},0,\theta\right)}{\partial\left(w^{m}_{i}\tilde{h}^{m,1}_{i}\right)}\cdot\frac{d\tilde{h}^{m,1}_{i}}{d\theta}w^{m}_{i}-\frac{dT\left(w^{m}_{i}\tilde{h}^{m,1}_{i},0,\theta\right)}{d\theta}\Bigg{]}=\\\
\frac{d\tilde{c}^{1}_{i}}{d\theta}+\left(1-\tau^{m}_{i}\right)w^{m}_{i}\left(\frac{d\tilde{h}^{m,2}_{i}}{d\theta}-\frac{d\tilde{h}^{m,1}_{i}}{d\theta}\right)+\left(1-\tau^{f}_{i}\right)w^{f}_{i}\frac{d\tilde{h}^{f}_{i}}{d\theta}-\frac{da_{i}}{d\theta}\left[w^{m}_{i}\left(\tilde{h}^{m,2}_{i}-\tilde{h}^{m,1}_{i}\right)+w^{f}_{i}\tilde{h}^{f}_{i}\right]$
(A.5)

where I denote the reform-induced change in the effective participation tax
rate as

$\frac{da_{i}}{d\theta}\equiv\frac{dT\left(w^{m}_{i}\tilde{h}^{m,2}_{i},w^{f}_{i}\tilde{h}^{f}_{i},\theta\right)/d\theta-
dT\left(w^{m}_{i}\tilde{h}^{m,1}_{i},0,\theta\right)/d\theta}{w^{m}_{i}\left(\tilde{h}^{m,2}_{i}-\tilde{h}^{m,1}_{i}\right)+w^{f}_{i}\tilde{h}^{f}_{i}}$

Next, I plug (A.5) into (A.4) and use the first-order conditions from the
expenditure minimization problem for dual-earner and single-earner couples to
get

$\frac{d\tilde{q}_{i}}{d\theta}=\left(\frac{d\tilde{c}^{1}_{i}}{d\theta}-\left(1-\tau^{m}_{i}\right)w^{m}_{i}\frac{d\tilde{h}^{m,1}_{i}}{d\theta}\right)\left(\frac{\partial
v_{i}\left(\tilde{c}^{2}_{i},\tilde{h}^{m,2}_{i},\tilde{h}^{f}_{i}\right)}{\partial\tilde{c}^{2}_{i}}-\frac{\partial
v_{i}\left(\tilde{c}^{1}_{i},\tilde{h}^{m,1}_{i},0\right)}{\partial\tilde{c}^{1}_{i}}\right)-\\\
\frac{\partial
v_{i}\left(\tilde{c}^{2}_{i},\tilde{h}^{m,2}_{i},\tilde{h}^{f}_{i}\right)}{\partial\tilde{c}^{2}_{i}}\cdot\frac{da_{i}}{d\theta}\left[w^{m}_{i}\left(\tilde{h}^{m,2}_{i}-\tilde{h}^{m,1}_{i}\right)+w^{f}_{i}\tilde{h}^{f}_{i}\right]$

Notice that the first multiplier in the first term is equal to zero, and
therefore I obtain

$\frac{d\tilde{q}_{i}}{d\theta}=-\frac{\partial
v_{i}\left(\tilde{c}^{2}_{i},\tilde{h}^{m,2}_{i},\tilde{h}^{f}_{i}\right)}{\partial\tilde{c}^{2}_{i}}\cdot\frac{da_{i}}{d\theta}\left[w^{m}_{i}\left(\tilde{h}^{m,2}_{i}-\tilde{h}^{m,1}_{i}\right)+w^{f}_{i}\tilde{h}^{f}_{i}\right]$
(A.6)

Plugging (A.6) into $dF_{i}\left(\tilde{q}_{i}\right)/d\theta=\left(\partial
F_{i}\left(\tilde{q}_{i}\right)/\partial\tilde{q}_{i}\right)\cdot\left(d\tilde{q}_{i}/d\theta\right)$,
I get

$\frac{dF_{i}\left(\tilde{q}_{i}\right)}{d\theta}=-\frac{\partial
F_{i}\left(\tilde{q}_{i}\right)}{\partial\tilde{q}_{i}}\cdot\frac{\partial
v_{i}\left(\tilde{c}^{2}_{i},\tilde{h}^{m,2}_{i},\tilde{h}^{f}_{i}\right)}{\partial\tilde{c}^{2}_{i}}\cdot\frac{da_{i}}{d\theta}\left[w^{m}_{i}\left(\tilde{h}^{m,2}_{i}-\tilde{h}^{m,1}_{i}\right)+w^{f}_{i}\tilde{h}^{f}_{i}\right]$
(A.7)

From the definition of the compensated participation elasticity, (19),

$\eta_{i}=\frac{\partial
F_{i}\left(\tilde{q}_{i}\right)}{\partial\tilde{q}_{i}}\cdot\frac{d\tilde{q}_{i}}{d(1-a_{i})}\cdot\frac{1-a_{i}}{F_{i}\left(\tilde{q}_{i}\right)}=-\frac{\partial
F_{i}\left(\tilde{q}_{i}\right)}{\partial\tilde{q}_{i}}\cdot\frac{d\tilde{q}_{i}}{da_{i}}\cdot\frac{1-a_{i}}{F_{i}\left(\tilde{q}_{i}\right)}$
(A.8)

To get the expression for $d\tilde{q}_{i}/da_{i}$, I use (A.6):

$\frac{d\tilde{q}_{i}}{da_{i}}=\frac{d\tilde{q}_{i}/d\theta}{da_{i}/d\theta}=-\frac{\partial
v_{i}\left(\tilde{c}^{2}_{i},\tilde{h}^{m,2}_{i},\tilde{h}^{f}_{i}\right)}{\partial\tilde{c}^{2}_{i}}\left[w^{m}_{i}\left(\tilde{h}^{m,2}_{i}-\tilde{h}^{m,1}_{i}\right)+w^{f}_{i}\tilde{h}^{f}_{i}\right]$
(A.9)

Plugging (A.9) into (A.8), we obtain:

$\eta_{i}=\frac{\partial
F_{i}\left(\tilde{q}_{i}\right)}{\partial\tilde{q}_{i}}\cdot\frac{1-a_{i}}{F_{i}\left(\tilde{q}_{i}\right)}\cdot\frac{\partial
v_{i}\left(\tilde{c}^{2}_{i},\tilde{h}^{m,2}_{i},\tilde{h}^{f}_{i}\right)}{\partial\tilde{c}^{2}_{i}}\left[w^{m}_{i}\left(\tilde{h}^{m,2}_{i}-\tilde{h}^{m,1}_{i}\right)+w^{f}_{i}\tilde{h}^{f}_{i}\right]$
(A.10)

Next, I plug (A.10) into (A.7), and obtain

$\frac{dF_{i}\left(\tilde{q}_{i}\right)}{d\theta}=-\frac{F_{i}\left(\tilde{q}_{i}\right)}{1-a_{i}}\cdot\frac{da_{i}}{d\theta}\eta_{i}$
(A.11)

Finally, I plug (A.1)-(A.3) and (A.11) into (28), and obtain

$\frac{dD}{d\theta}=\sum_{i=1}^{N}\Bigg{[}\left(\varepsilon^{m,2}_{i}\frac{\tau^{m}_{i}}{1-\tau^{m}_{i}}\cdot\frac{d\tau^{m}_{i}}{d\theta}+\varepsilon^{mf}_{i}\frac{\tau^{m}_{i}}{1-\tau^{f}_{i}}\cdot\frac{d\tau^{f}_{i}}{d\theta}\right)F_{i}\left(\tilde{q}_{i}\right)w^{m}_{i}\tilde{h}^{m,2}_{i}+\\\
\varepsilon^{m,1}_{i}\frac{\tau^{m}_{i}}{1-\tau^{m}_{i}}\cdot\frac{d\tau^{m}_{i}}{d\theta}\left(1-F_{i}\left(\tilde{q}_{i}\right)\right)w^{m}_{i}\tilde{h}^{m,1}_{i}+\left(\varepsilon^{f}_{i}\frac{\tau^{f}_{i}}{1-\tau^{f}_{i}}\cdot\frac{d\tau^{f}_{i}}{d\theta}+\varepsilon^{fm}_{i}\frac{\tau^{f}_{i}}{1-\tau^{m}_{i}}\cdot\frac{d\tau^{m}_{i}}{d\theta}\right)F_{i}\left(\tilde{q}_{i}\right)w^{f}_{i}\tilde{h}^{f}_{i}+\\\
\eta_{i}\frac{a_{i}}{1-a_{i}}\cdot\frac{da_{i}}{d\theta}F_{i}\left(\tilde{q}_{i}\right)\left[w^{m}_{i}\left(\tilde{h}^{m,2}_{i}-\tilde{h}^{m,1}_{i}\right)+w^{f}_{i}\tilde{h}^{f}_{i}\right]\Bigg{]}$
(A.12)

Dividing this expression by aggregate labor income, (29), I obtain equation
(30). This completes the proof of Proposition 1. $\blacksquare$

### Proof of Proposition 2 (Efficiency Loss under Nonlinear Taxation of
Couples)

This proof extends Blomquist and Simula (2019) to the framework with couples.
The utility maximization problem of couple is given by

$\max_{c,y_{m},y_{f}}v\left(c,y_{m},y_{f},\upsilon_{m},\upsilon_{f}\right)$
(B.1)
$\text{s.t.}~{}~{}~{}~{}~{}c=y_{m}+y_{f}-T\left(y_{m},y_{f},\theta\right)+I$
(B.2)

where $y_{m}$ and $y_{f}$ are taxable incomes of a male and a female,
$\upsilon_{m}$ and $\upsilon_{f}$ are individual specific preference
parameters of a male and a female, and $I$ is lump-sum non-taxable income. As
before, $\theta$ is a treatment parameter that captures the tax policy
reforms. Preference parameters
$\boldsymbol{\upsilon}=\left(\upsilon_{m},\upsilon_{f}\right)$ are jointly
drawn from continuous distribution $\varGamma$. Denote the solution to this
problem by $c\left(\upsilon_{m},\upsilon_{f},\theta,I\right)$,
$y_{m}\left(\upsilon_{m},\upsilon_{f},\theta,I\right)$, and
$y_{f}\left(\upsilon_{m},\upsilon_{f},\theta,I\right)$.

Next, to get the compensated functions, I turn to the expenditure minimization
problem that is given by

$\min_{c,y_{m},y_{f}}c-y_{m}-y_{f}+T\left(y_{m},y_{f},\theta\right)-I$ (B.3)
$\text{s.t.}~{}~{}~{}~{}~{}v\left(c,y_{m},y_{f},\upsilon_{m},\upsilon_{f}\right)\geq\bar{U}$
(B.4)

where $\bar{U}$ is a fixed level of utility. The solution delivers compensated
functions $\tilde{c}\left(\upsilon_{m},\upsilon_{f},\theta,\bar{U}\right)$,
$\tilde{y}_{m}\left(\upsilon_{m},\upsilon_{f},\theta,\bar{U}\right)$, and
$\tilde{y}_{f}\left(\upsilon_{m},\upsilon_{f},\theta,\bar{U}\right)$.

Note that, at the optimum, the following equality holds:

$v\left(\tilde{c},\tilde{y}_{m},\tilde{y}_{f},\upsilon_{m},\upsilon_{f}\right)=\bar{U}$
(B.5)

Using the compensated functions, I write the expenditure function as

$E\left(\upsilon_{m},\upsilon_{f},\theta,\bar{U}\right)=\tilde{c}-\tilde{y}_{m}-\tilde{y}_{f}+T\left(\tilde{y}_{m},\tilde{y}_{f},\theta\right)-I$
(B.6)

Next, set $\bar{U}$ to be the indirect utility delivered by the utility
maximization problem (B.1)-(B.2). Then the solutions to the expenditure
minimization problem and utility maximization problem coincide.

Consistent with (24), I use the measure of excess burden based on the
equivalent variation:

$D\left(\upsilon_{m},\upsilon_{f},\theta,\bar{U}\right)=E\left(\upsilon_{m},\upsilon_{f},\theta,\bar{U}\right)-E\left(\upsilon_{m},\upsilon_{f},0,\bar{U}\right)-R\left(\upsilon_{m},\upsilon_{f},\theta,\bar{U}\right)$
(B.7)

The tax revenue function
$R\left(\upsilon_{m},\upsilon_{f},\theta,\bar{U}\right)$ is given by

$R\left(\upsilon_{m},\upsilon_{f},\theta,\bar{U}\right)=T\left(\tilde{y}_{m},\tilde{y}_{f},\theta\right)$
(B.8)

Aggregate deadweight loss from a tax and transfer system $\theta$ is obtained
by integrating excess burdens over all couples:

$D=\int
D\left(\upsilon_{m},\upsilon_{f},\theta,\bar{U}\right)d\varGamma\left(\upsilon_{m},\upsilon_{f}\right)$
(B.9)

Next, plugging (B.8) into (B.7), I obtain

$D=\int\left[E\left(\upsilon_{m},\upsilon_{f},\theta,\bar{U}\right)-T\left(\tilde{y}_{m},\tilde{y}_{f},\theta\right)-E\left(\upsilon_{m},\upsilon_{f},0,\bar{U}\right)\right]d\varGamma\left(\upsilon_{m},\upsilon_{f}\right)$
(B.10)

Applying the envelope theorem to (B.6), I obtain
$dE\left(\upsilon_{m},\upsilon_{f},\theta,\bar{U}\right)/d\theta=\partial
T\left(\tilde{y}_{m},\tilde{y}_{f},\theta\right)/\partial\theta$, i.e. the
small tax reform affects the expenditure function only through mechanical
revenue effect. Differentiating aggregate excess burden (B.10) with respect to
parameter $\theta$ and using the result from the envelope theorem, I get

$\frac{dD}{d\theta}=-\int\Bigg{[}\frac{\partial
T\left(\tilde{y}_{m},\tilde{y}_{f},\theta\right)}{\partial\tilde{y}_{m}}\cdot\frac{d\tilde{y}_{m}}{d\theta}+\frac{\partial
T\left(\tilde{y}_{m},\tilde{y}_{f},\theta\right)}{\partial\tilde{y}_{f}}\cdot\frac{d\tilde{y}_{f}}{d\theta}\Bigg{]}d\varGamma\left(\upsilon_{m},\upsilon_{f}\right)$
(B.11)

Next, I rewrite the expression for marginal deadweight loss in terms of the
curvatures of the indifference curve and the tax function. Consider a small
reform captured by $d\theta\approx 0$. In the expenditure minimization
problem, assume that constraint (B.4) is binding:

$v\left(c,y_{m},y_{f},\upsilon_{m},\upsilon_{f}\right)=\bar{U}$ (B.12)

Define function
$c=\psi\left(y_{m},y_{f},\upsilon_{m},\upsilon_{f},\bar{U}\right)$, and plug
it into the objective (B.3):

$\min_{y_{m},y_{f}}\psi\left(y_{m},y_{f},\upsilon_{m},\upsilon_{f},\bar{U}\right)-y_{m}-y_{f}+T\left(y_{m},y_{f},\theta\right)-I$
(B.13)

From (B.13), I get the following first-order conditions:

$\frac{\partial\psi\left(\tilde{y}_{m},\tilde{y}_{f},\upsilon_{m},\upsilon_{f},\bar{U}\right)}{\partial\tilde{y}_{j}}-1+\frac{\partial
T\left(\tilde{y}_{m},\tilde{y}_{f},\theta\right)}{\partial\tilde{y}_{j}}=0,~{}~{}~{}~{}~{}~{}~{}j=m,f$
(B.14)

Differentiating (B.14) with respect to policy parameter $\theta$, I obtain

$\frac{\partial^{2}\psi\left(\tilde{y}_{m},\tilde{y}_{f},\upsilon_{m},\upsilon_{f},\bar{U}\right)}{\partial\left(\tilde{y}_{m}\right)^{2}}\frac{d\tilde{y}_{m}}{d\theta}+\frac{\partial^{2}\psi\left(\tilde{y}_{m},\tilde{y}_{f},\upsilon_{m},\upsilon_{f},\bar{U}\right)}{\partial\tilde{y}_{m}\partial\tilde{y}_{f}}\frac{d\tilde{y}_{f}}{d\theta}+\frac{\partial^{2}T\left(\tilde{y}_{m},\tilde{y}_{f},\theta\right)}{\partial\tilde{y}_{m}\partial\theta}+\\\
\frac{\partial^{2}T\left(\tilde{y}_{m},\tilde{y}_{f},\theta\right)}{\partial\left(\tilde{y}_{m}\right)^{2}}\frac{d\tilde{y}_{m}}{d\theta}+\frac{\partial^{2}T\left(\tilde{y}_{m},\tilde{y}_{f},\theta\right)}{\partial\tilde{y}_{m}\partial\tilde{y}_{f}}\frac{d\tilde{y}_{f}}{d\theta}=0$
(B.15)
$\frac{\partial^{2}\psi\left(\tilde{y}_{m},\tilde{y}_{f},\upsilon_{m},\upsilon_{f},\bar{U}\right)}{\partial\left(\tilde{y}_{f}\right)^{2}}\frac{d\tilde{y}_{f}}{d\theta}+\frac{\partial^{2}\psi\left(\tilde{y}_{m},\tilde{y}_{f},\upsilon_{m},\upsilon_{f},\bar{U}\right)}{\partial\tilde{y}_{m}\partial\tilde{y}_{f}}\frac{d\tilde{y}_{m}}{d\theta}+\frac{\partial^{2}T\left(\tilde{y}_{m},\tilde{y}_{f},\theta\right)}{\partial\tilde{y}_{f}\partial\theta}+\\\
\frac{\partial^{2}T\left(\tilde{y}_{m},\tilde{y}_{f},\theta\right)}{\partial\left(\tilde{y}_{f}\right)^{2}}\frac{d\tilde{y}_{f}}{d\theta}+\frac{\partial^{2}T\left(\tilde{y}_{m},\tilde{y}_{f},\theta\right)}{\partial\tilde{y}_{m}\partial\tilde{y}_{f}}\frac{d\tilde{y}_{m}}{d\theta}=0$
(B.16)

Denote
$\psi^{\prime\prime}_{ij}\equiv\partial^{2}\psi(\cdot)/\partial\tilde{y}_{i}\partial\tilde{y}_{j}$,
$T^{\prime\prime}_{ij}\equiv\partial^{2}T(\cdot)/\partial\tilde{y}_{i}\partial\tilde{y}_{j}$,
and
$T^{\prime\prime}_{i\theta}\equiv\partial^{2}T(\cdot)/\partial\tilde{y}_{i}\partial\theta$.
Solving for $d\tilde{y}_{m}/d\theta$ and $d\tilde{y}_{f}/d\theta$ from
equations (B.15) and (B.16), I get the expressions in terms of curvatures of
the indifference curve and tax function:

$\frac{d\tilde{y}_{m}}{d\theta}=\frac{\left(\psi^{\prime\prime}_{mf}+T^{\prime\prime}_{mf}\right)T^{\prime\prime}_{f\theta}-\left(\psi^{\prime\prime}_{ff}+T^{\prime\prime}_{ff}\right)T^{\prime\prime}_{m\theta}}{\left(\psi^{\prime\prime}_{mm}+T^{\prime\prime}_{mm}\right)\left(\psi^{\prime\prime}_{ff}+T^{\prime\prime}_{ff}\right)-\left(\psi^{\prime\prime}_{mf}+T^{\prime\prime}_{mf}\right)^{2}}$
(B.17)
$\frac{d\tilde{y}_{f}}{d\theta}=\frac{\left(\psi^{\prime\prime}_{mf}+T^{\prime\prime}_{mf}\right)T^{\prime\prime}_{m\theta}-\left(\psi^{\prime\prime}_{mm}+T^{\prime\prime}_{mm}\right)T^{\prime\prime}_{f\theta}}{\left(\psi^{\prime\prime}_{mm}+T^{\prime\prime}_{mm}\right)\left(\psi^{\prime\prime}_{ff}+T^{\prime\prime}_{ff}\right)-\left(\psi^{\prime\prime}_{mf}+T^{\prime\prime}_{mf}\right)^{2}}$
(B.18)

Denoting $T^{\prime}_{i}\equiv\partial T(\cdot)/\partial y_{i}$ and plugging
(B.17) and (B.18) into (B.11), I obtain the expression for reform-induced
efficiency loss under nonlinear taxation of couples:

$\frac{dD}{d\theta}=-\int\Bigg{[}\frac{T^{\prime}_{m}\left[\left(\psi^{\prime\prime}_{mf}+T^{\prime\prime}_{mf}\right)T^{\prime\prime}_{f\theta}-\left(\psi^{\prime\prime}_{ff}+T^{\prime\prime}_{ff}\right)T^{\prime\prime}_{m\theta}\right]}{\left(\psi^{\prime\prime}_{mm}+T^{\prime\prime}_{mm}\right)\left(\psi^{\prime\prime}_{ff}+T^{\prime\prime}_{ff}\right)-\left(\psi^{\prime\prime}_{mf}+T^{\prime\prime}_{mf}\right)^{2}}+\\\
\frac{T^{\prime}_{f}\left[\left(\psi^{\prime\prime}_{mf}+T^{\prime\prime}_{mf}\right)T^{\prime\prime}_{m\theta}-\left(\psi^{\prime\prime}_{mm}+T^{\prime\prime}_{mm}\right)T^{\prime\prime}_{f\theta}\right]}{\left(\psi^{\prime\prime}_{mm}+T^{\prime\prime}_{mm}\right)\left(\psi^{\prime\prime}_{ff}+T^{\prime\prime}_{ff}\right)-\left(\psi^{\prime\prime}_{mf}+T^{\prime\prime}_{mf}\right)^{2}}\Bigg{]}d\varGamma\left(\upsilon_{m},\upsilon_{f}\right)$
(B.19)

Linearized Tax Function

Given $\left(\upsilon_{m},\upsilon_{f},\theta,I\right)$, a couple solve the
problem (B.1)-(B.2), and I denote the solution by
$c^{*}=c\left(\upsilon_{m},\upsilon_{f},\theta,I\right)$,
$y_{m}^{*}=y_{m}\left(\upsilon_{m},\upsilon_{f},\theta,I\right)$, and
$y_{f}^{*}=y_{f}\left(\upsilon_{m},\upsilon_{f},\theta,I\right)$. I linearize
the tax system, so that now it is described by proportional tax rates
$\tau_{m}=\partial T\left(y_{m},y_{f},\theta\right)/\partial y_{m}$ and
$\tau_{f}=\partial T\left(y_{m},y_{f},\theta\right)/\partial y_{f}$ and a
lump-sum component. Namely,

$T^{L}\left(y_{m},y_{f},\tau_{m},\tau_{f}\right)=\tau_{m}(\theta)y_{m}+\tau_{f}(\theta)y_{f}+T^{*}$
(B.20)

This linearized tax system delivers $\left(c^{*},y_{m}^{*},y_{f}^{*}\right)$
as a solution if I set
$T^{*}=y_{m}^{*}+y_{f}^{*}-c^{*}-\tau_{m}y^{*}_{m}+\tau_{f}y^{*}_{f}+I$. To
simplify notation, I omit explicit dependence of $\tau_{m}$ and $\tau_{f}$ on
parameter $\theta$. Under linearized tax system, the couple’s budget
constraint is given by

$c=\left(1-\tau_{m}\right)y_{m}+\left(1-\tau_{f}\right)y_{f}+I-T^{*}$ (B.21)

Denote the solution to the problem with a linearized tax function, given by
(B.1) and (B.21), by
$c^{L}\left(\upsilon_{m},\upsilon_{f},\tau_{m},\tau_{f},I-T^{*}\right)$,
$y_{m}^{L}\left(\upsilon_{m},\upsilon_{f},\tau_{m},\tau_{f},I-T^{*}\right)$,
and
$y_{f}^{L}\left(\upsilon_{m},\upsilon_{f},\tau_{m},\tau_{f},I-T^{*}\right)$.
By construction, it coincides with $\left(c^{*},y_{m}^{*},y_{f}^{*}\right)$.

Next, solving the expenditure minimization problem with a linearized tax
function, I get the compensated functions
$\tilde{c}^{L}\left(\upsilon_{m},\upsilon_{f},\tau_{m},\tau_{f},\bar{U}\right)$,
$\tilde{y}_{m}^{L}\left(\upsilon_{m},\upsilon_{f},\tau_{m},\tau_{f},\bar{U}\right)$,
and
$\tilde{y}_{f}^{L}\left(\upsilon_{m},\upsilon_{f},\tau_{m},\tau_{f},\bar{U}\right)$.
Using the expression for marginal deadweight loss (B.11), I obtain

$\frac{dD^{L}}{d\theta}=-\int\Bigg{[}\frac{\partial
T^{L}\left(\tilde{y}^{L}_{m},\tilde{y}^{L}_{f},\tau_{m},\tau_{f}\right)}{\partial\tilde{y}^{L}_{m}}\cdot\frac{d\tilde{y}^{L}_{m}}{d\theta}+\frac{\partial
T^{L}\left(\tilde{y}^{L}_{m},\tilde{y}^{L}_{f},\tau_{m},\tau_{f}\right)}{\partial\tilde{y}^{L}_{f}}\cdot\frac{d\tilde{y}^{L}_{f}}{d\theta}\Bigg{]}d\varGamma\left(\upsilon_{m},\upsilon_{f}\right)$
(B.22)

Now, I rewrite this expression for reform-induced efficiency loss in terms of
the curvatures of the indifference curve and tax function. Under a linearized
tax function, it is true that
$\left(T^{L}\right)^{\prime\prime}_{ij}\equiv\partial^{2}T^{L}(\cdot)/\partial\tilde{y}_{i}^{L}\partial\tilde{y}_{j}^{L}=0$.
Furthermore, since, by construction,
$\left(T^{L}\right)^{\prime}_{i}=T^{\prime}_{i}$, then it is also true that
$\left(T^{L}\right)^{\prime\prime}_{i\theta}=T^{\prime\prime}_{i\theta}$.
Therefore, using these results in (B.17) and (B.18), I obtain

$\frac{d\tilde{y}_{m}^{L}}{d\theta}=\frac{\psi^{\prime\prime}_{mf}T^{\prime\prime}_{f\theta}-\psi^{\prime\prime}_{ff}T^{\prime\prime}_{m\theta}}{\psi^{\prime\prime}_{mm}\psi^{\prime\prime}_{ff}-\left(\psi^{\prime\prime}_{mf}\right)^{2}}$
(B.23)
$\frac{d\tilde{y}_{f}^{L}}{d\theta}=\frac{\psi^{\prime\prime}_{mf}T^{\prime\prime}_{m\theta}-\psi^{\prime\prime}_{mm}T^{\prime\prime}_{f\theta}}{\psi^{\prime\prime}_{mm}\psi^{\prime\prime}_{ff}-\left(\psi^{\prime\prime}_{mf}\right)^{2}}$
(B.24)

Plugging (B.23) and (B.24) into (B.22), I obtain the expression for marginal
deadweight loss under linearized tax function:

$\frac{dD^{L}}{d\theta}=-\int\Bigg{[}\frac{T^{\prime}_{m}\left(\psi^{\prime\prime}_{mf}T^{\prime\prime}_{f\theta}-\psi^{\prime\prime}_{ff}T^{\prime\prime}_{m\theta}\right)}{\psi^{\prime\prime}_{mm}\psi^{\prime\prime}_{ff}-\left(\psi^{\prime\prime}_{mf}\right)^{2}}+\frac{T^{\prime}_{f}\left(\psi^{\prime\prime}_{mf}T^{\prime\prime}_{m\theta}-\psi^{\prime\prime}_{mm}T^{\prime\prime}_{f\theta}\right)}{\psi^{\prime\prime}_{mm}\psi^{\prime\prime}_{ff}-\left(\psi^{\prime\prime}_{mf}\right)^{2}}\Bigg{]}d\varGamma\left(\upsilon_{m},\upsilon_{f}\right)$
(B.25)

This completes the proof of Proposition 2. $\blacksquare$

### Proof of Proposition C.1 (Efficiency Loss with HSV Tax Function)

Before proving Proposition 3, first, I state and prove Proposition C.1 that
characterizes the expressions for marginal deadweight losses under joint and
separate taxation of spousal incomes.