
%\subsection{Balance as a function of savings} \label{sec:appx_eqns}

%The Roth version of Equation \ref{eqn:brok_bal} is:
%\begin{equation} \label{eqn:roth_bal}
%	B^{ROTH}_t = \sum_{J=0}^{\infty} S^{ROTH}_{t-J} \Pi_{j=1}^{J} (1+r_{t-j})
%\end{equation}
%The traditional version of Equation \ref{eqn:brok_bal} is:
%\begin{equation} \label{eqn:trad_bal}
%	B^{TRAD}_t = \sum_{J=0}^{\infty} \frac{1}{1-\tau^Y_{t-J}-\sigma_{i,t-J}} S^{TRAD}_{t-J} \Pi_{j=1}^{J} (1+r_{t-j})
%\end{equation}


\subsection{Comparing the present value of different retirement accounts: the more general case} \label{sec: pv_general}

In the main text, we consider a closed form comparison of the present value of Roth and traditional accounts under a simplifying assumption that contributions are made in exactly one year and distributions are made in exactly one year. In this appendix, we consider the more general case when contributions are made at any arbitrary schedule (through $t^*-1$) and distributions are made at any arbitrary schedule (in $t^*$ or later).

The present value of a Roth account is equal to the balance at the end of the period preceding retirement. This occurs because the discount rate is equal to the rate of return within the account. Thus, continuing to assume $\sigma_{it} = 0$ for exposition, combining the budget constraint and the accumulation equations gives
\begin{equation} \label{eqn:roth_summary}
    A_{i}^{ROTH} = B_{it^*}^{ROTH} = \sum_{s=\underline{t}}^{t^*-1} S_{is}R^{s,t^*} \text{.}
\end{equation}

In fact, when comparing traditional and Roth accounts, it is more convenient to express distributions and contributions in pre-tax terms. Let lowercase letters denote these pre-tax equivalents -- i.e., $s_{it} = \frac{S_{it}}{1-\tau_{it}^Y}$  and $d_{it} = \frac{D_{it}}{1-\tau_{it}^Y}$. For Roth accounts, then, the present value can be written as
\begin{equation} \label{eqn:roth_summary_trad}
    A_{i}^{ROTH} = B_{it^*}^{ROTH} = \sum_{s=\underline{t}}^{t^*-1} s_{is}(1-\tau_{it}^Y) \times R^{s,t^*} \text{.}
\end{equation}



For traditional accounts, the accumulation equation becomes
\begin{equation} \label{eqn: trad_accum_pre}
    B_{it^*}^{TRAD} = \sum_{s=\underline{t}}^{t^*-1} s_{is} R^{s,t^*} \text{.}
\end{equation}
Likewise, the decumulation equation for traditional accounts (i.e., the budget constraint) becomes
\begin{equation} \label{eqn: trad_decum_pre}
    B_{it^*}^{TRAD} = \sum_{s=t^*}^{\overline{t}} \frac{d_{is}}{R^{t^*,s}} 
\end{equation}
and the present value equation becomes
\begin{equation} \label{eqn: pv_trad}
    A_i^{TRAD} = \sum_{s=t^*}^{\overline{t}} \frac{d_{it}(1-\tau_{it}^Y)}{R^{t^*,s}} \text{.}
\end{equation}
We can combine Equations \ref{eqn: trad_decum_pre} and \ref{eqn: pv_trad} to express the present value as 
\begin{equation} \label{eqn:trad_summary}
    A_{i}^{TRAD} = \big(1-\bar{\tau}_i^{Y,POST}\big) B_{it^*}^{TRAD} \text{,}
\end{equation}
where $\bar{\tau}_i^{Y,POST}$ is the weighted average of $\tau_{it}^Y$ in retirement, weighted by the present value of pre-tax distributions, $\frac{d_{is}}{R^{t^*,s}}$.


Furthermore, combining this with  Equation \ref{eqn: trad_decum_pre}, we have
\begin{equation} \label{eqn: trad_pretax}
    A_{i}^{TRAD} = \Big(1-\bar{\tau}_i^{Y,POST}\Big) \sum_{s=\underbar{t}}^{t^*-1} s_{is} R^{s,t^*}
\end{equation}

Comparing Equations \ref{eqn:roth_summary_trad} and \ref{eqn: trad_pretax}, we see that traditional accounts lead to higher $A$ if and only if $\bar{\tau}_i^{Y,POST}$ is less than the weighted average of $\tau_{it}^Y$ during working age, where the weights are $s_{it}R^{t,t^*}$. Thus, as in the simpler case, when tax rates tend to be lower in retirement than in working age, traditional accounts are favored relative to Roth accounts.

%$\bar{\tau}^{POST}_i$ is smaller than $\bar{\tau}^{PRE}_i$, where the latter is the appropriately weighted average tax rate during the accumulation phase.\footnote{The weights are the at-retirement value of net-of-tax savings, (i.e., $S_{it}R^{t,t^*}$).} 

% \sum_{s=\underbar{t}}^{t^*-1}  \frac{1-\bar{\tau}_i^{POST}}{1-\tau_{it}} \times S_{it} R^{s,t^*} \text{. }

%Finally, consider the present value of a brokerage account. Brokerage accounts can be most readily compared to Roth accounts, especially with the simplification that $\sigma_{it} = 0$. Because the inside build-up in brokerage accounts is taxed, $B_{it^*}^{BROK} < B_{it^*}^{ROTH}$ for the same saving path. Furthermore, while the present value of a Roth account is equal to its balance, the present value of a brokerage account is \textit{less} than its balance, as the rate of return within the account is lower than the discount rate. In sum,
%\begin{equation} \label{eqn:brok_summary}
%    A_{i}^{BROK} < B_{it^*}^{BROK} = \sum_{t=\underbar{t}}^{t^*-1} S_{it} \tilde{R}_i^{t,t^*} \text{,}
%\end{equation}
%where the difference between $B_{it^*}^{BROK}$ and $A_{i}^{BROK}$ is a complicated function of tax rates in retirement.



\subsection{Constructing the panel from 1960 through 1998} \label{sec: pre_panel}

% interpolation

% Wage split

% extrapolation pre-1969

% itemized deductions, including SALT

% secondary filer in 1969

% state tax calculation? maybe that goes later.

In this section, we provide more details on the construction of the panel prior to 1999. In particular, our panel needs to include wages of each individual (which is used in the savings imputation discussed in the main text) as well as enough information to compute federal and state taxes.

From 1996-1998, we observe the usual set of variables from Form 1040 and its schedules, but not from any information returns. Our only task is to estimate the wages earned by the individual in our sample. For single individuals, this is trivial -- wages are simply equal to the wages reported on Form 1040. For married individuals, we impute the split of wages between husband and wife,\footnote{Same-sex marriage was not legal in this time period.} as discussed below.

Prior to 1996, we have access to the population of Forms 1040 in roughly 5-year intervals -- specifically, 1969, 1974, 1979, 1984, 1994, and 1995. However, these years are ``narrower'' datasets, meaning that we do not observe as much detail about income sources or deductions. In particular, in these data we can consistently observe filing status, the identity of spouse (in 1974 and later), wages, dividends (in 1974 and later), AGI, dependents, and state. We assume that all AGI other than wages and dividends is taxed as ordinary income. 

\textbf{Wage imputation.}
While we do observe wages on Form 1040, we do not observe the split between spouses. We impute the wage split for each observation using data from the Current Population Survey Annual Social and Economic Supplement (CPS ASEC), retrieved from IPUMS CPS \citep{ipums_cps}. In particular, separately in each year of the CPS, we regress the husband's share of wages on a set of covariates (log total wages, dummies for number of dependents, a quadratic in the wife's age, a quadratic in the husband's age, and the interaction of wife's and husband's age) that we also observe in the tax data; we apply these coefficients in the tax data to impute the husband's share to each tax observation.

\textbf{Itemized deduction imputation.}
We do not observe itemized deductions. We impute itemized deductions by assuming that itemized deductions are some fraction $\gamma_t$ of AGI. We calibrate $\gamma_t$ to match historical aggregates. In particular, for a given guess of $\gamma_t$, we estimate aggregate ``total deductions'' (that is, the larger of itemized or standard deductions for a given taxpayer) using a random sample of the full population Form 1040 from that year. We then compare that 
to the analogous aggregate amount as calculated by the IRS Statistics of Income (SOI) division in Historical Table 7.\footnote{\url{https://www.irs.gov/statistics/soi-tax-stats-historical-table-7}} We then find the $\gamma_t$ that best matches this aggregate.

Additionally, we must impute the deduction for real estate taxes, as this deduction figures into the post-TCJA \$10,000 limitation on the deduction for state and local taxes. We assume that real estate taxes are the same share of total itemized deductions as they were in 2016, as reported in aggregate SOI statistics.\footnote{Individual Complete Report (Publication 1304), Table 2.1.}




\textbf{Additional 1969 imputations.}
The 1969 file has two additional complications. First, the 1969 dataset does not identify the secondary filer. As secondary filers are overwhelmingly wives (which is especially true in this time period), we proceed as follows. For all wives in 1974 that are not primary filers in 1969, we look to see if their 1974 husband filed a joint return in 1969. If so, we assume that the husband was married to the same wife in 1969, and use that 1969 joint return for that observation.

Second, the 1969 file does not contain information on dividends. We assume that the dividend share of AGI in 1969 for each observation was equal to their dividend share of AGI in 1974.

\textbf{Interpolation.}
We linearly interpolate all dollar variables between data years. We additionally linearly interpolate the number of dependents, though we round to the nearest integer. In the case that filing status or state of residence changes from one data year to the next, we assume that the change occurs as close to the midpoint of the two data years as possible.

\textbf{Extrapolation prior to 1969.}
We extrapolate backwards to 1960 as follows. First, in the 1970 CPS ASEC (which reflects income earned in 1969), we estimate mean wages by sex and age, denoted $y_{sa}$, for sex $s$ and age $a$. We then scale all dollar amounts from 1960-1968 by the ratio of mean wages in that year to mean wages in 1969. Specifically, for a given variable with value $x_{i,1969}$ in 1969, $x_{it}$ is set to equal 
\begin{equation}
x_{it} = x_{i,1969}\times\frac{y_{s(i),a(it)}}{y_{s(i),a(i,1969)}} \times \frac{P_t}{P_{1969}} \text{, }
\end{equation}
where $\frac{P_{1969}}{P_t}$ represents price inflation from $t$ to $1969$.

Based on Census Table MS-2\footnote{\url{https://www.census.gov/data/tables/time-series/demo/families/marital.html}}, we assume that all men who were married in 1969 got married at the earlier of age 23 or 1969 and all women who were married in 1969 got married at the earlier of age 20 or 1969. If the individual has dependents in 1969, we linearly interpolate between the initial year of marriage (as imputed above) and 1969, assuming zero dependents in the initial year of marriage, and then round to the nearest integer. 



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                           %
%       Additional calculation details      %
%                                           %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Additional computation details} \label{sec: appendix_misc}

\textbf{The half-year assumption.}
The law of motion in Section \ref{sec:conceptual} implies that balances are beginning-of-year balances and that contributions and distributions are made immediately after the year begins. Empirically, we expect contributions and distributions to be made more smoothly throughout the year. %(See [Ramnath et al short JPubE paper] for evidence in favor of this in the context of distributions.) 
As an approximation to this supposition, we assume throughout our empirical exercise that distributions and contributions are in fact made halfway during the year (i.e., on July 1).

This requires the following adjustments and interpretations to the equations in Section \ref{sec:conceptual}. First, we continue to define balances as of the beginning of the period. Second, the cumulative rate of return from $t<t^*$ to $t^*$ is interpreted as the cumulative rate of return from July 1 of $t$ to the beginning of $t^*$. In the tax-preferred case, this is equal to $\sqrt{1+r_t}\times \Pi_{s=t+1}^{t^*-1} (1+r_{s})$ -- i.e., the component of the return attributable to year $t$ is only the six month return during $t$.\footnote{We do not attempt to measure actual returns more finely than annually.} Third, the cumulative rate of return from $t^*$ to $t>t^*$ is interpreted as the cumulative rate of return from the beginning of $t^*$ to July 1 of $t$. In the tax-preferred case, this is again equal to $\sqrt{1+r_t}\times \Pi_{s=t^*}^{t-1} (1+r_{s})$. Finally, we adjust the simulated path of distributions such that a certain fraction of the middle-of-year (pre-distribution) balance is distributed in a given year, rather than the beginning-of-year balance. That is, Equation \ref{eq: dist} becomes
\begin{equation}\label{eq: dist_adj}
 d_{it} = \frac{1}{(\overline{t}+1-t)^\gamma} \times \Big(B_{it} (1+r_t)^{(\frac{1}{2})}\Big)
\end{equation}


\textbf{Adjusting observed Roth balances to their traditional equivalent.}
For the sake of imputing savings based on observed traditional IRA balances, we must adjust observed balances in Roth accounts to their traditional equivalent. We do so by assuming that saving in Roth accounts is a fixed fraction of total saving from the introduction of Roth IRAs in 1998 through 2002. In combination with the assumptions below, this assumption implies that we can convert Roth balances to their traditional equivalent by inflating the observed Roth balances by the ratio of a weighted mean of $\frac{1}{1-\tau_{it}-\sigma_{it}^{TRAD}}$ to a weighted mean of $\frac{1}{1-\sigma_{it}^{ROTH}}$, where the weights are equal to $f_{it}$ (as defined in section \ref{sec: panel}) times net-of-tax wages.

\textbf{Adjusting for the introduction of traditional IRAs in 1975.}
In our simulations, we allow savings to occur as early as 1960. However, traditional IRAs did not exist until 1975. Thus, both for the purpose of (a) imputing the path of savings and (b) simulating counterfactual wealth under the contemporaneous law counterfactuals, we make the following assumption. We assume that individuals saved in a taxable brokerage account from 1960-1974, then used those funds to transfer the net-of-tax equivalent to an IRA from 1975 to 1980. 

Formally, this requires us to make a distinction between savings $S_{it}$, which represents the amount of consumption deferred in a given year, and $\widehat{S_{it}}$, the net-of-tax contributions made to a given account. The former is the economic primitive -- i.e., that is what we assume is equal to $\alpha_i \times f_{it} w_{it}$ -- while the latter is what is relevant for the law of motion of balance within the account. Therefore, in these settings, we proceed empirically by replacing $S_{it}$ with $\widehat{S_{it}}$ in the equations in Section \ref{sec:conceptual}.

From 1981 onward (and in all years under the 2019 law counterfactual), $\widehat{S_{it}}$ is simply equal to $S_{it}$. But, while imputing savings and under the contempaneous law counterfactuals, we now have $\widehat{S_{it}} = 0$ for $t < 1975$ and, for $1975 \leq t \leq 1980$, 
\begin{equation} \label{eqn: n_vs_s}
    \widehat{S_{it}} = S_{it} + Z_{it} \text{,}
\end{equation}
where $Z_{it}$ is the amount of net-of-tax wealth transferred from the taxable account to the retirement account. We assume that
\begin{equation} \label{eqn: z}
    Z_{it} = \frac{1}{1980+1-t} B_{it}^{BROK} \times \big(1+r_t(1-\tau^K_{it})\big)^{(\frac{1}{2})}\text{,}
\end{equation}
That is, a certain fraction (starting at $\frac{1}{6}$ and increasing to $\frac{1}{1}$) of the mid-year balance in the taxable brokerage account is transferred to the retirement account. In the absence of investment returns from 1975 to 1980, this assumption would imply a constant transfer of wealth between 1975 and 1980.


\textbf{Computing tax rates on contributions.}
In section \ref{sec:conceptual} we have approximated $\tau_{it}^Y$ and $\sigma_{it}$ as constants (within $i$ and $t$); however, the relevant tax rates and saver's credit rates are the \textit{average} rates with respect to the amount of the contribution. The average tax rate and marginal tax rate may differ if a contribution causes an individual to change tax brackets. Furthermore, the saver's credit average rate may differ from the saver's credit marginal rate because the saver's credit applies only to the first \$2,000 of contributions per individual. As a result, both of these rates are, in principle, a non-linear function of the contribution amount.

In particular, the more general formula for traditional contributions ($x_{it}^{TRAD}$) as a function of net-of-tax savings $\widehat{S_{it}}$ is
\begin{equation} \label{eqn: saving_trad}
    x_{it}^{TRAD} = \frac{\widehat{S_{it}}}{1 - \tau_{it}^Y(x_{it}^{TRAD}) - \sigma_{it}(x_{it}^{TRAD})} \text{, }
\end{equation}
where $\tau_{it}^Y(\cdot)$ and $\sigma_{it}(\cdot)$ are the average rates (i.e., the change in tax divided by $x_{it}^{TRAD}$), which are a function of $x_{it}^{TRAD}$.

We apply an iterative algorithm to find the value of $x_{it}^{TRAD}$ that makes (\ref{eqn: saving_trad}) hold. In very rare cases, this algorithm does not converge -- e.g., if an individual is near an AGI cutoff for the saver's credit. In these cases, we take the average $\tau_{it} + \sigma_{it}$ from two final iterations and treat this as the applicable tax rate. 

In the Roth case, we only perform one step of the iteration to compute $\sigma_{it}$, for computational reasons. In particular, we assume that contributions are equal to $\widehat{S_{it}}$ and compute $\sigma_{it}$ taking that contribution as given. 



\textbf{Decomposing the difference in retirement tax rates}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                       %
%       Begin validation appendix       %
%                                       %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Additional analyses}

\subsection{Benchmarking imputation accuracy} \label{sec: appendix_impute}

In the accumulation phase, there are two main shortcomings of the data that we must overcome. First, we observe earnings and other tax-relevant items only in the certain years (1969, 1974, 1979, 1984, 1989, and 1994 onward). Thus, we must interpolate between those years, as discussed above.  Second, we do not observe saving during almost all of working life, so we must impute that as well. In this appendix, we use an auxiliary dataset to explore the extent to which such imputation errors may alter our conclusions.

The most critical computation task in our empirical exercise is to calculate, for each individual, a weighted average marginal tax rate (WAMTR) that applies to contributions and one that applies to distributions. For contributions – which is what these imputations concern -- the relevant weights are the traditional contributions in a given year multiplied by the cumulative return from that year to retirement. The interpolation of tax items affects (mostly) the marginal tax rate in a given year and the imputation of savings affects (mostly) the weights. 

Using more recent working-age data, we can study how these imputations affect the WAMTR. In particular, we can compute the WAMTR for each individual under four cases, defined by the 2-by-2 interaction of (actual vs. imputed tax items) and (actual vs. imputed savings). 

We implement this as follows. First, we extract a auxiliary panel of working-age individuals from 2005 to 2015. Specifically, we define our population of interest to be those individuals who satisfy all of the following conditions: (1) they made any traditional or Roth contribution (DC or IRA) at any point between 2005 and 2015, (2) they had positive wages in any of 2005, 2010, or 2015, (3) they were  in the 1961-1980 cohorts (so that they were between age 25 and 54 at all points between 2005 and 2015), and (4) they were alive through the end of 2015.

We then take a 0.1\% random sample of this population, and form a panel from 2005 to 2015 out of these individuals. In each year, we observe relevant tax items and IRA and DC contributions. We construct and append four versions of this panel, corresponding to each of the four cases. When imputing tax items, we replace 2006-2009 (2011-2014) data with data linearly interpolated from 2005 and 2010 (2010 and 2015). When imputing saving, we follow an analogy to the methodology described in section \ref{sec: panel}. Specifically, we compute a notional ``balance'' entering 2016, based on actual contributions from 2005-2015, multiplied by the appropriate cumulative return. Then, we impute savings in 2005-2015 as if we observed that balance but not any actual savings amounts, i.e. assuming $S_{it} = \alpha_i g(age_{it}) w_{it}$ as in section \ref{sec: panel} and requiring that the contributions (grossed up by the appropriate cumulative return) sums to the notional balance.

Appendix Table \ref{tab: imputation_benchmark} presents our estimates for the average WAMTR under each of the four cases. The difference in average WAMTR between the truth and the other three cases is less than 1.5 percentage points in magnitude. These differences are modest in magnitude relative to the difference in marginal tax rate between the accumulation phase and decumulation phase, i.e. from Table \ref{tab: building_blocks}. Furthermore, imputing savings tends to increase it somewhat while interpolating tax items tends to decrease it; these imputation errors offset, meaning that the gap between the actual data and the fully imputed data is only about 0.5 percentage points. We interpret the body of this evidence as suggesting that the imputation details are not driving our results.





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                           %
%       Sensitivity and extensions          %
%                                           %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Sensitivity and extensions} \label{sec: appendix_sens}




\subsubsection{Alternative strategies} \label{sec: appendix_alt_strat}
 In Table \ref{tab: altstrats}, we examine several simple rules that workers might use to choose between saving in traditional and Roth accounts in a given year. Recall that the conventional advice to workers is to save in a Roth account if the worker's retirement-age marginal tax rate will exceed the worker's working-age marginal tax rate. Forecasting the retirement-age marginal tax rate is difficult. %\cite{brown2017tax} argue persuasively that there is an advantage to diversifying across traditional and Roth accounts within a calendar year to hedge marginal tax rate risk. 
For computational tractability, we restrict attention to strategies that allocate 100 percent of IRA savings within a calendar year to one account type, traditional or Roth.

The strategies we examine are based on stylized facts about wage paths and the distribution of marginal tax rates. The stylized fact about wage paths is that real wages tend to increase with age, and the simple strategy inspired by this stylized fact is for workers to save in a Roth account if they are age 30 or younger, and to save in a traditional account if they are age 31 or older. The stylized fact about the distribution of marginal tax rates is that Federal taxes have a graduated rate structure, and the simple strategy inspired by this stylized fact is for workers to save in a Roth account if their current-year federal marginal tax rate is in the 40th percentile or below -- this corresponds to 12 percent in the inflation-adjusted 2019 law counterfactual and 25 percent in the contemporaneous law counterfactual. In other words, this strategy asks workers to compare their current income to the current income of other workers and does not require a projection of future investment returns or future tax rates.

% numbers updated 1/10/2025
Table \ref{tab: altstrats} shows that, under contemporaneous law, any of these alternative strategies would lead to very similar outcomes as a traditional-only strategy. We interpret this as another indication of the importance of the 1986, 2001, and 2003 tax changes in determining retirement outcomes for this cohort. Under inflation-adjusted 2019 law, the strategy of contributing to a Roth account when the marginal tax rate is low (and traditional when not low) makes a slightly larger difference -- it increases retirement wealth by 2 to 4 percent relative to contributing to only Roth or only traditional accounts.


% We denote the latter as $\sigma_{it}^{TRAD}$ and $\sigma_{it}^{ROTH}$, reflecting the fact that they may (in principle) differ depending on whether the contribution is a Roth or traditional contribution.

\subsubsection{Alternative assumptions}
Appendix Table \ref{tab: alt_assumptions} reports the estimated tax shield under alternative assumptions about contributions, distributions, and investment portfolios. Columns (1)-(3) study contemporaneous law; while columns (4)-(6) study inflation-adjusted law. Under the contemporaneous law counterfactual, the supremacy of traditional accounts remains robust, so we focus our discussion in this section on how these alternate assumptions affect the inflation-adjusted 2019 law counterfactual.

% numbers updated 1/10/2025
First, we evaluate sensitivity to assumptions about \textit{contribution timing}. Recall that the baseline assumption about contributions is that contributions as a share of earnings follow the average path shown in Appendix Figure \ref{fig: contr_path}. To assess the sensitivity of the estimated tax shields to the baseline assumption about contribution timing, Table \ref{tab: alt_assumptions} presents results assuming (1) contributions as a share of earnings were constant and (2) contributions were made entirely within the final ten years of working life.\footnote{Specifically, we assume that contributions as a share of earnings increases linearly from zero over the ten final years of working life.} Relative to the baseline case, the former assumption places contributions earlier in working age, while the latter assumption places contributions later in working age. In both cases the level of contributions is calibrated to match the observed IRA balance in 2003. Contributions earlier in working life increase the absolute shield relative to taxable accounts, as the larger cumulative returns benefit more from the tax-free inside build-up. Furthermore, contributions later in working life, when marginal tax rates tend to be higher, generally increases the advantage of traditional accounts over Roth accounts. When contributions are made closer to the end of working life in the inflation-adjusted 2019 law counterfactual, 65\% of the cohort could have financed more consumption with a traditional account than a Roth account. This compares to 53\% in the baseline case and 51\% when contributions are made earlier in working life.

Next, we evaluate sensitivity to assumptions about \textit{distribution timing}. The distribution timing affects our calculations primarily through its effect on the average tax rate on the balance in the traditional account, and to a lesser extent by changing the magnitude of the tax-free inside build-up during retirement itself.\footnote{Put differently, the assumptions about distribution timing do not affect the effective wealth held in Roth accounts, which is mechanically equal to the balance in the Roth account.} In the baseline, we assume that distributions follow the observed distributions of traditional IRAs, scaled up or down to satisfy the budget constraint. In the alternative, we assume that ditributions in time $t$ are given by
\begin{equation} \label{eq: dist}
    d_{it}^{k} = \frac{1}{(\overline{t}_i+1-t)^\gamma} \times B_{it}^k \text{,}
\end{equation}
where $\overline{t}_i$ is the earlier of 2022 or year of death.

This assumption states that a certain fraction $\frac{1}{(\overline{t}+1-t)^\gamma}$ of the  balance $B_{it}^k$ is distributed each year. In the ``intermediate'' case, we set $\gamma = 1$, which implies that pre-tax distributions are equal in present value in all years in retirement. We note that this path of distributions mechanically satisfies the budget constraint, as it implies that any remaining balance at $\overline{t}$ is distributed in that year. In Table \ref{tab: alt_assumptions}, we vary $\gamma$ such that distributions tend to happen earlier ($\gamma=\frac{2}{3}$) or later ($\gamma = \frac{4}{3}$) in retirement. 

In all cases, smooth distributions cause traditional accounts to be noticeably more favored relative to Roth accounts. As discussed in \ref{sec: building_blocks}, this occurs both because true distributions are lumpy (i.e., the IRA distribution itself causes an increase in the average tax rate that applies to that income) and because individuals tend to take distributions exactly when their tax rate tends to be higher for other reasons. 

Changing the timing of these smooth assumptions has relatively more modest effects. Making distributions later in retirement (relative to the intermediate case) increases the tax shield of both tax-preferred accounts, while doing so earlier in retirement decreases it. This reflects the change in tax-free build-up during retirement. Additionally, both earlier and later distributions generally favor Roth accounts relative to the intermediate case: distributions tend to be more concentrated under either alternative assumption, which tends to increase the marginal tax rate on distributions. 

%If distributions were earlier in retirement, we estimate a tax shield of 0.27 for traditional accounts and 0.25 for Roth accounts, and 51 percent of the cohort would have financed more retirement consumption with a traditional account. If distributions were later in retirement, we estimate a tax shield of 0.36 for traditional accounts and 0.28 for Roth accounts, and 68 percent of the cohort would have financed more retirement consumption with a traditional account. Using as much of the observed path of traditional distributions as possible, we estimate a tax shield of 0.34 for traditional accounts and 0.29 for Roth accounts, and 56 percent of the cohort would have financed more retirement consumption with a traditional account.

% numbers updated 1/10/2025
Finally, we assess the sensitivity of the estimated tax shields to the baseline assumption about \textit{portfolio composition}. Varying portfolio composition is assumed to affect only how returns are taxed in a taxable account; portfolio composition is assumed not to affect pre-tax returns, and thus it has no impact on the relative ranking of traditional and Roth accounts.
Recall that the baseline assumption about \textit{portfolio composition} is that 50 percent of investment returns are long term capital gains, 25 percent are qualified dividends, and 25 percent are taxable interest. Table \ref{tab: alt_assumptions} presents results assuming (1) 100 percent of investment returns are taxable interest, (2) 50 percent of investment returns are dividends and 50 percent are long term capital gains, and (3) 100 percent of investment returns are long term capital gains. If investment returns were entirely taxable interest, tax shields are much larger, reflecting the fact that interest income is taxed on accrual at ordinary rates in taxable brokerage accounts: we estimate tax shields that are 50 to 85 percent larger than the baseline case, depending on the account and the assumptions about law. If investment returns were entirely long term capital gains---which have the lowest accrual-equivalent tax rate in brokerage accounts---tax shields are 25 to 35 percent lower than the baseline assumptions. 



\subsubsection{Saver's credit} \label{sec: appendix_savers}

Table \ref{tab: savers} examines the impact of the saver's credit on estimated tax shields. To isolate the impact of the saver's credit, it estimates average retirement wealth in several scenarios that all assume constant inflation-adjusted 2019 law other than the saver's credit. The first scenario assumes no saver's credit existed during the working lives of the 2003 retirement cohort. The second scenario assumes the saver's credit existed as it did in 2019 during the working lives of the 2003 cohort, i.e. a non-refundable credit with tiered match rates; this is the baseline inflation-adjusted 2019 law counterfactual. The third scenario assumes the saver's credit existed as passed by the SECURE 2.0 Act of 2022 during the working lives of the 2003 cohort, i.e. a refundable credit with a smooth phaseout that is scheduled to take effect in tax year 2027. The SECURE 2.0 saver's credit works differently than the 2019 saver's credit in that the credit is generally deposited into a traditional account,\footnote{We do not model any of the exceptions to this general rule, such as when the credit is under \$100.} rather than reducing tax liability.

%The 2019 saver's credit and the SECURE 2.0 saver's credit both augment the impact of after-tax savings on retirement account balances.

% numbers updated 8/7/2024
We estimate that the 2019 saver's credit would have added around 5 percentage points to the tax shield of both traditional and Roth accounts. The SECURE 2.0 saver's credit adds an incremental 4 percentage points to the tax shield of each type of account.

% numbers updated 8/7/2024

We expect the saver's credit to be more favorable towards savings in traditional accounts than Roth accounts because the match rate is effectively higher on traditional contributions. The match is expressed as a share of dollars of contributions, but because traditional contributions are pre-tax and Roth contributions are post-tax, a dollar of traditional contributions reflects less effective saving than a dollar of Roth contributions. However, we find that this theoretical advantage has a relatively small effect empirically: the SECURE 2.0 saver's credit increases the share of individuals preferring traditional accounts by approximately 2.9 percentage points relative to no saver's credit.


