In this paper, we study the relative tax advantages of traditional and Roth retirement accounts in the United States by examining the retrospective tax benefits of each account type for the 2003 cohort of retirees. We use the actual history of marginal tax rates at the individual level to estimate that the average tax shields of traditional and Roth retirement accounts‒the excess value of retirement consumption financed by those accounts relative to brokerage accounts‒were 68 percent and 47 percent, respectively. Traditional accounts were better for this cohort largely due to tax cuts in 2001 and 2003. Under a counterfactual with inflation-adjusted 2019 law, the average tax shields for traditional and Roth accounts would have been much closer (33% and 30%, respectively), and nearly half of savers would have achieved higher retirement by saving in Roth accounts rather than in traditional accounts. This is the first paper to apply the history of marginal tax rates‒informed by administrative income records‒to compare Roth and traditional accounts for a particular cohort of retirees.

A linear equation $E$ is said to be sparse if there is $c\gt 0$ so that every subset of $[n]$ of size $n^{1-c}$ contains a solution of $E$ in distinct integers. The problem of characterising the sparse equations, first raised by Ruzsa in the 90s, is one of the most important open problems in additive combinatorics. We say that $E$ in $k$ variables is abundant if every subset of $[n]$ of size $\varepsilon n$ contains at least $\text{poly}(\varepsilon )\cdot n^{k-1}$ solutions of $E$. It is clear that every abundant $E$ is sparse, and Girão, Hurley, Illingworth, and Michel asked if the converse implication also holds. In this note, we show that this is the case for every $E$ in four variables. We further discuss a generalisation of this problem which applies to all linear equations.